Extremal Problems in Discrete Geometry and Spectral Graph Theory

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Doctoral Thesis

Author(s): Balla, Igor

Publication Date: 2019

Permanent Link: https://doi.org/10.3929/ethz-b-000366468

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ETH Library Albert Einstein 1905 Spectral GraphTheory Discrete Geometryand Extremal Problemsin Igor Balla Diss. ETH No.

26048

igor balla EXTREMALPROBLEMSINDISCRETEGEOMETRYAND SPECTRALGRAPHTHEORY

diss. eth no. 26048

EXTREMALPROBLEMSINDISCRETEGEOMETRYAND SPECTRALGRAPHTHEORY

A dissertation submitted to attain the degree of doctor of sciences of eth zurich (Dr. sc. ETH Zurich)

presented by igor balla M. sc. New York University born on 26 January 1990 citizen of the United States of America

accepted on the recommendation of Prof. Dr. B. Sudakov, examiner Prof. Dr. Y. Zhao, co-examiner

We study several extremal problems lying at the intersection of combinatorics and linear algebra. Many of the problems we will be considering can be thought of as associating a matrix with a corresponding graph and then using the combinatorial properties of this graph in order to bound natural parameters of the matrix, such as the largest eigenvalue or the rank. Our proofs make use of extremal and probabilistic combinatorial methods, together with linear algebraic techniques and inequalities, as well as the polynomial method. One class of problems we will consider involve equiangular lines and spherical codes. The problem of finding the maximal number of equiangular lines in d-dimensional Euclidean space has a long history going back 70 years, with several interesting questions arising as a by-product. In this thesis, we make significant progress on some of these questions. In particular, motivated by a question of Lem- mens and Seidel from 1973, we prove that for every fixed angle θ and sufficiently large n there are at most 2n − 2 lines in Rn with common angle θ, where this bound 1 is achieved if and only if θ = arccos 3 . We also provide various extensions of these results to the more general setting of spherical codes, as well as proposing and studying generalizations to k-dimensional subspaces. The other class of problems we will consider can be seen as spectral generalizations of well-known combinatorial problems and involve the Lovász ϑ-function ϑ(G), as well as the min-rank mrF(G) and minimum semidefinite rank msr(G) of a graph G. We study the maximum of ϑ G and the minimum of msr(G) over all H- free graphs G on n vertices. For H being a triangle, these quantities are understood and correspond to questions of Lovász and Erd˝os.We obtain results for various other graphs H, in particular showing a connection between the Turán number of H and the maximum of ϑ G over all H-free graphs G, for certain bipartite graphs H. We also study the min-rank of the Erd˝os-Rényirandom graph G(n, p) over an arbitrary field F, showing that for p fixed, it is on the order of n/ log n with high probability. This answers a question of Knuth for the minimum semidefinite rank of G(n, p), and applies to other geometric representations of G(n, p) as well.

ZUSAMMENFASSUNG

Wir untersuchen einige Extremalprobleme im Grenzbereich der Kombinatorik und der linearen Algebra. Viele der Probleme, die wir behandeln werden, können wie folgt betrachtet werden; man nutzt die kombinatorischen Eigenschaften des zu ei- ner Matrix assoziierten Graphen um natürliche Parameter der Matrix, wie dem grössten Eigenwert oder dem Rang, zu beschränken. Unsere Beweise verwenden Methoden der extremalen und probabilistischen Kombinatorik, zusammen mit Techniken und Ungleichungen der linearen Algebra, sowie die Polynomialmetho- de. Eine Klasse von Problemen die wir behandeln beinhalten gleichwinklige Linien und sphärische Kodierungen. Das Problem die maximale Anzahl gleichwinkliger Linien im d-dimensionalen Euklidischen Raum zu finden, hat eine lange Geschich- te, die 70 Jahre zurück reicht, mit einigen interessanten Fragen, die als Nebenpro- dukt auftauchen. In dieser These machen wir signifikanten Fortschritt in einigen dieser Fragen. Insbesondere, motiviert durch eine Frage von Lemmens und Seidel von 1973, beweisen wir für einen fixen Winkel θ und n genügend gross, dass es höchstens 2n − 2 Linien in Rn gibt, so dass der Winkel zwischen je zwei Linien θ 1 beträgt und dass diese obere Grenze genau dann erreicht wird wenn θ = arccos 3 . Wir geben auch einige Erweiterungen dieser Resultate auf allgemeinere Situatio- nen für sphärische Kodierungen. Ausserdem schlagen wir Verallgemeinerungen für k-dimensionale Unterräume vor und studieren diese. Die andere Klasse von Problemen die wir behandeln, kann man als spektra- le Verallgemeinerungen bekannter kombinatorischer Probleme betrachten und sie beinhalten die Lovász ϑ-Funktion ϑ(G), sowie den min-Rang mrF(G) und den minimalen semidefiniten Rang msr(G) eines Graphen G. Wir untersuchen das Ma- ximum von mrF(G) und das Minimum von msr(G) über allen H-freien Graphen G mit n Ecken. Diese Grössen sind verstanden falls H ein Dreieck ist und entsprechen Fragen von Lovász and Erd˝os.Wir erhalten Resultate für diverse andere Graphen H, insbesondere zeigen wir eine Verbindung zwischen der Turánnummer von H und dem Maximum von ϑ G über alle H-freien Graphen G, für gewisse bipartite Graphen H. Wir untersuchen auch den min-Rang des Erd˝os-RényiZufallsgraphen G(n, p) über allen beliebigen Körpern F, und zeigen, dass er, für ein fixes p, mit grosser Wahrscheinlichkeit im Bereich n/ log n liegt. Dies beantwortet die Frage von Knuth über das Minimum des semidefiniten Ranges von G(n, p), und ist auch auf andere geometrische Repräsentationen von G(n, p) übertragbar.

vii

ACKNOWLEDGEMENTS

I would firstly like to thank my advisor, Benny Sudakov, for all of the guidance and wisdom he shared with me, whether it be mathematical or otherwise. I will always be grateful for the opportunities he provided me and for putting up with my less than perfect work ethic. I would also like to thank Oded Regev for introducing me to Benny and I would like to thank Peter Brooks, Joseph Stern, and John Mackey for being passionate and outstanding teachers, without whom I would not have chosen to pursue mathematics. I would also like to thank the rest of “Team Benny”, especially Adam, Alexey, Dano, Felix, Jan, John, Matija, Matthew, Nina, Omri, Pedro, Rajko, Shoham, and Tuan, for being excellent colleagues, officemates, lunch conversationists, and friends of mine over the past 4 years. In addition, I would like to thank Davide Spriano for being an awesome officemate and I would like to thank Alex Puttick and Alessan- dro Sisto for being especially cool. I would like to thank Berit Singer for helping with the German translation of the abstract and for being a friend. I am deeply grateful to my family for their love and support, in spite of my lack of calling. The amount that they have helped me throughout my life is more than I am capable of putting into words. I would also like to thank all of my friends back in New York, especially Andrew Burban, Arthur Podlesniy, and Pavel Korecky, for staying in touch with me despite the distance and timezone difference. I would like to give an enormous thank you to Toro Adeyemi for being a special person in my life for over 7 years, especially since many of them were spent apart. I would like to give a special shout-out to my friends in the Swiss SSBM community for being so inviting and chill. I would also like to thank Jim Bern, Victor Gelehrter, and Noemi Shavit for being wonderful neighbors and friends. Lastly, I would like to thank all of my coauthors, Noga Alon, Bela Bollobás, Felix Dräxler, Tom Eccles, Lior Gishboliner, Peter Keevash, Shoham Letzter, Adva Mond, Frank Mousset, Alexey Pokrovskiy, and Benny Sudakov, for the successful research projects, as well as Boris Bukh and all of the other people with whom I have had interesting mathematical discussions. Finally, I would like to thank Yufei Zhao for agreeing to be my co-examiner. The chapters in this thesis are based on papers I completed during my PhD. Chapter 2 is based on the paper “Equiangular Lines and Spherical Codes in Eu- clidean Space” which is joint work with Felix Dräxler, Peter Keevash, and Benny Sudakov. Chapter 3 is based on the paper “Equiangular subspaces in Euclidean spaces” which is joint work with Benny Sudakov. Chapter 4 is based on the paper “Orthonormal representations of H-free graphs” which is joint work with Shoham Letzter and Benny Sudakov. Chapter 5 is based on the paper “The Minrank of Random Graphs over Arbitrary Fields” which is joint work with Noga Alon, Lior Gishboliner, Adva Mond, and Frank Mousset.

CONTENTS

1 introduction1

2 equiangular lines and spherical codes3 2.1 Introduction 3 2.1.1 Notation 6 2.2 Equiangular lines 6 2.2.1 Orthogonal projections 7 2.2.2 Spectral techniques 11 2.2.3 Proof of the main result 14 2.3 Spherical codes 16 2.4 A construction 20 2.5 Lines with many angles and related spherical codes 22 2.5.1 A general bound when β is ﬁxed 22 2.5.2 An asymptotically tight bound when β, α1,..., αk are ﬁxed 24 2.6 Concluding remarks 30

3 equiangular subspaces 33 3.1 Introduction 33 3.2 Angle distances 34 3.3 Other distances 37 3.4 Concluding remarks 39

4 orthonormal representations of H-free graphs 41 4.1 Introduction 41 4.1.1 A geometric problem of Lovász 42 4.1.2 Almost orthogonal vectors 44 4.2 Minimum semideﬁnite rank for H-free graphs 46 4.3 Lovász ϑ-function for H-free graphs 48 4.4 Concluding remarks 56

5 the min-rank of random graphs 57 5.1 Introduction 57 5.2 Preliminaries 58 5.3 The min-rank of random graphs 61 5.4 Geometric representations of random graphs 62 5.5 Concluding remarks 65 a appendix 69 bibliography 73

1 INTRODUCTION

I like mathematics because it is not human and has nothing particular to do with this planet or with the whole accidental universe - because, like Spinoza’s God, it won’t love us in return. — Bertrand Russell

The connections between graph theory and linear algebra are natural and nu- merous, and have been studied from various points of view in the combinatorics, geometry, applied mathematics, and computer science communities. As mentioned in the abstract, many of the problems we will be studying can be formalized by associating a graph with a matrix and then using the structure of the graph in order to say something about some parameter of the matrix, such as its rank or its largest eigenvalue. We will see that many interesting and important problems have this form, in which case we can use both combinatorial and linear-algebraic tools to study them. A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle, and the question of determining the maximum size of an equiangular set of lines in Rn is considered to be one of the founding problems of algebraic graph theory, see e.g. [42, p. 249]. If we choose a unit vector along each line, we can observe that since the angle between any pair of lines is the same, the inner product of any pair of vectors will be ±α for some constant α. Therefore, we may define a graph whose vertices are the vectors and there is an edge between u and v if hu, vi = +α. Moreover, we may consider the Gram matrix of all pairwise inner products between the vectors, and observe that our graph encodes which entries in this matrix are +α and which are −α. Since this matrix is positive semidefinite and has rank n, we may equivalently think of the problem of maximizing the number of equiangular lines as finding the minimum dimension of a positive semidefinite matrix whose diagonal entries are 1 and whose off-diagonal entries are ±α. It is known that the maximum size of an equiangular set of lines in Rn is on the order of n2. In Chapter 2, we use Ramsey’s theorem together with spectral methods to show that for every fixed angle θ and sufficiently large n there are at most 2n − 2 lines in Rn with common angle θ. Moreover, this bound is achieved if 1 1 and only if θ = arccos 3 . Indeed, we show that for all θ 6= arccos 3 and sufficiently large n, the number of equiangular lines is at most 1.93n. We also show that for any set of k fixed angles, one can find at most O(nk) lines in Rn having these angles. This bound, conjectured by Bukh, substantially improves the estimate of Delsarte, Goethals and Seidel from 1975. Moreover, we also prove extensions of these results to more general spherical codes.

1 2 introduction

In Chapter 3, we generalize the concept of equiangular lines to k-dimensional subspaces. To do so, we first discuss natural ways of defining the angle between k-dimensional subspaces and then we correspondingly study the maximum size of an equiangular set of k-dimensional subspaces in Rn. We show that for a class of possible definitions of angle, which we call angle distances, there can be no more than O(n2k) equiangular k-dimensional subspaces in Rn, generalizing the bound for equiangular lines, and extending and improving a result of Blokhuis for equiangular planes. For k fixed, we also use the known construction of Ω(n2) equiangular lines in order to construct a set of Ω(n2k) k-dimensional subspaces in Rn which are equiangular with respect to any proper angle distance. For a graph G, an orthonormal representation is an assignment of a unit vector to each vertex such that if a pair of vertices are not adjacent, then the inner product between their corresponding vectors must be 0. Given such a representation, we may consider the Gram matrix of all pairwise inner products between the vectors. Then the minimum over all orthonormal representations of the rank of the Gram matrix is defined to be the minimum semidefinite rank msr(G) of G and the maximum over all orthonormal representations of the largest eigenvalue of the Gram matrix is defined to be the Lovász ϑ-function ϑ(G) of G. Note that these parameters are well-studied, partly due to the fact that they provide bounds for the Shannon capacity of a graph. In Chapter 4, we consider the maximum of ϑ G and the minimum of msr(G) over all H-free graphs G on n vertices. For H being a triangle, these correspond to well-known questions of Lovász and Erd˝os,and we obtain results for more general H. Interestingly, for certain kinds of bipartite graphs H, we show a connection between the Turán number of H and the maximum of ϑ G over H-free graphs G. For H being a cycle of length t, we show that the minimum of msr(G) over all H-free graphs G is exactly dn/(t − 1)e. This generalizes Rosenfeld’s result for almost orthogonal vectors (t = 3), and improves a result of Pudlák. The minrank of a graph G on the set of vertices [n] over a field F is the minimum possible rank of a matrix M ∈ Fn×n with nonzero diagonal entries such that Mi,j = 0 whenever i and j are distinct nonadjacent vertices of G. This notion generalizes the minimum semidefinite rank discussed above and is related to important problems in circuit complexity and information theory. In Chapter 5, we obtain tight bounds for the typical minrank of the Erd˝os-Rényirandom graph G(n, p) over any finite or infinite field, showing that for every field F = F(n) and every p = p(n) satisfying n−1 ≤ p ≤ 1 − n−0.99, the minrank of G(n, p) over F is ( n log(1/p) ) Θ log n with high probability. The result for the real field settles a problem raised by Knuth in 1994, and we discuss applications to other geometric representations of random graphs. The proof is based on a recent argument of Golovnev, Regev, and Weinstein, who proved the above result for finite fields of size at most nO(1), with tools from linear algebra, including an estimate of Rónyai, Babai, and Ganapathy for the number of zero-patterns of a sequence of polynomials. 2 EQUIANGULARLINESANDSPHERICALCODES

2.1 introduction

A set of lines through the origin in n-dimensional Euclidean space is called equiangular if any pair of lines defines the same angle. Equiangular sets of lines appear naturally in various areas of mathematics. In elliptic geometry, they correspond to equilateral sets of points, or, in other words, to regular simplexes. These simplexes were first studied 70 years ago [45], since the existence of large regular simplexes leads to high congruence orders of elliptic spaces, see [13, 46, 66]. In frame theory, so-called Grassmannian frames “are characterized by the property that the frame elements have minimal cross-correlation among a given class of frames” [52]. It turns out that optimal Grassmannian frames are equiangular; hence searching for equiangular sets of lines is closely related to searching for optimal Grassmannian frames, see [52]. In the theory of polytopes, the convex hull of the points of intersection of an equiangular set of lines with the unit sphere is a spherical polytope of some kind of regularity, see [23]. It is therefore a natural question to determine the maximum cardinality N(n) of an equiangular set of lines in Rn. As stated in the introduction, this is also considered to be one of the founding problems of algebraic graph theory, see e.g. [42, p. 249]. While it is easy to see that N(2) ≤ 3 and that the three diagonals of a regular hexagon achieve this bound, matters already become more difficult in 3 dimensions. This problem was first studied by Haantjes [45] in 1948, who showed that N(3) = N(4) = 6 and that an optimal configuration in 3 (and 4) dimensions is given by the 6 diagonals of a convex regular icosahedron. In 1966, van Lint and Seidel [66] formally posed the problem of determining N(n) for all positive integers n and furthermore showed that N(5) = 10, N(6) = 16 and N(7) ≥ 28. The general upper bound n+1 N(n) ≤ ( 2 ) (2.1) was established by Gerzon (see [64]). Let us outline his proof. Given an equiangular n set of m lines in R , one can choose a unit vector xi along the ith line to obtain D E vectors x1,..., xm satisfying xi, xj ∈ {−α, α} for i 6= j. Consider the family of | n+1 outer products xixi ; they live in the ( 2 )-dimensional space of symmetric n × n matrices, equipped with the inner product hA, Bi = tr(A|B). It is a routine 2 D | |E D E 2 calculation to verify that xixi , xjxj = xi, xj , which equals α if i 6= j and 1 otherwise. This family of matrices is therefore linearly independent, which implies n+1 m ≤ ( 2 ).

3 4 equiangular lines and spherical codes

In dimensions 2 and 3 this gives upper bounds of 3 and 6, respectively, matching the actual maxima. In R7, the above bound shows N(7) ≤ 28. This can be achieved by considering the set of all 28 permutations of the vector (1, 1, 1, 1, 1, 1, −3, −3), see [66, 81]. Indeed, one can verify that the dot product of any two distinct such vectors maximizingequals either −8 or 8, so that after normalizing the vectors to unit length this constitutes an equiangular set of lines. Since the sum of the coordinates of each such vector is 0, they all live in the same 7-dimensional subspace. It is also known that there is an equiangular set of 276 lines in R23, see e.g. [64], which again matches Gerzon’s bound. Strikingly, these four examples are the only known ones to match his bound [9]. In fact, for a long time it was even an open problem to determine whether n2 is the correct order of magnitude. In 2000, de Caen [20] constructed a set of 2(n + 1)2/9 equiangular lines in Rn for all n of the form 3 · 22t−1 − 1, so that we have N(n) ≥ Ω n2 .(2.2)

Subsequently, several other constructions of the same order were found [9, 41, 54]. For further progress on finding upper and lower bounds on N(n) see e.g. [9] and its references. 2 Interestingly, all√ the above examples of size Θ(n ) have a common angle on the order of arccos(1/ n). On the other hand, all known construction of equiangular lines with a fixed common angle have much smaller size. It is therefore natural to n consider the maximum number Nα(n) of equiangular lines in R with common angle arccos α, where α does not depend on dimension. This question was first raised by Lemmens and Seidel [64] in 1973, who showed that for sufficiently large n, N1/3(n) = 2n − 2 and also conjectured that N1/5(n) equals b3(n − 1)/2c. This conjecture was later confirmed by Neumaier [74], see also [41] for more details. Interest in the case where 1/α is an odd integer was due to a general result of Neumann [64, p. 498], who proved that if Nα(n) ≥ 2n, then 1/α is an odd integer. Despite active research on this problem, for many years these were the best results known. Recently, Bukh [16] made important progress by showing that O(1/α2) Nα(n) ≤ cαn, where cα = 2 is a large constant only depending on α. Our first main result completely resolves the question of maximizing Nα(n) over constant α. 1 We show that for sufficiently large n, Nα(n) is maximized at α = 3 . Theorem 2.1.1. Fix α ∈ (0, 1). For n sufficiently large relative to α, the maximum number n 1 of equiangular lines in R with angle arccos α is exactly 2n − 2 if α = 3 and at most 1.93n otherwise. A more general setting than that of equiangular lines is the framework of spherical L-codes, introduced in a seminal paper by Delsarte, Goethals and Seidel [26] in 1977 and extensively studied since. Definition 2.1.2. Let L be a subset of the interval [−1, 1). A finite non-empty set C of unit vectors in Euclidean space Rn is called a spherical L-code, or for short an L-code, if hx, yi ∈ L for any pair of distinct vectors x, y in C. 2.1 introduction 5

Note that if L = {−α, α}, then an L-code corresponds to a set of equiangular lines with common angle arccos α, where α ∈ [0, 1). For L = [−1, β], finding the maximum cardinality of an L-code is equivalent to the classical problem of finding 1 non-overlapping spherical caps of angular radius 2 arccos β; for β ≤ 0 exact formu- lae were obtained by Rankin [77]. Generalizing Gerzon’s result, Delsarte, Goethals and Seidel [25] obtained bounds on the cardinality of sets of lines having a prescribed number of angles. They proved that, for L = {−α1,..., −αk, α1,..., αk} and 2k α1,..., αk ∈ [0, 1), spherical L-codes have size at most O(n ). They subsequently extended this result to an upper bound of O(ns) on the size of an L-code when L has cardinality s, see [26]. A short proof of this estimate based on the polynomial method is due to Koornwinder [60]. Bukh [16] observed that, in some sense, the negative values of L pose less of a constraint on the size of L-codes than the positive ones, as long as they are sep- arated away from 0. Specifically, he proved that for L = [−1, −β] ∪ {α}, where β ∈ (0, 1) is fixed, the size of any L-code is at most linear in the dimension. Moti- vated by the above-mentioned work of Delsarte, Goethals and Seidel [25] he made the following conjecture.

Conjecture 2.1.3. Let β ∈ (0, 1) be ﬁxed and let α1,..., αk be any k reals. Then any n k spherical [−1, −β] ∪ {α1,..., αk}-code in R has size at most cβ,kn for some constant cβ,k depending only on β and k. We verify this conjecture in the following strong form.

Theorem 2.1.4. Let L = [−1, −β] ∪ {α1,..., αk} for some ﬁxed β ∈ (0, 1]. Then there ex- n k ists a constant cβ,k such that any spherical L-code in R has size at most cβ,kn . Moreover, if 0 ≤ α1 < ... < αk < 1 are also ﬁxed then such a code has size at most α 2k(k − 1)! 1 + 1 nk + o(nk). β

In particular, if α1,..., αk are ﬁxed this substantially improves the aforementioned bound of Delsarte, Goethals and Seidel [25, 26] from O(n2k) to O(nk). We furthermore show that the second statement of Theorem 2.1.4 is tight up to a constant factor. Theorem 2.1.5. Let n, k, r be positive integers and α ∈ (0, 1) with k and α being ﬁxed √ 1 p 1 and r ≤ n. Then there exist α2,..., αk, β = α1/r − O( log(n)/n) and a spherical n n+r L-code of size (1 + r)(k) in R with L = [−1, −β] ∪ {α1,..., αk}. This also resolves another question of Bukh, who asked whether the maximum n size√ of a spherical [−1, 0) ∪ {α}-code in R is linear in n. By choosing r on the order of n/ log(n), our construction demonstrates that this is not the case. The rest of this chapter is organized as follows. In Section 2.2 we give a construction of an equiangular set of 2n − 2 lines in Rn and prove Theorem 2.1.1. In Section 2.3 we prove a special case of Theorem 2.1.4, namely the case k = 1. We provide the construction which shows that our bounds are asymptotically tight in Section 4. In Section 2.5, we prove Theorem 2.1.4. The last section of the chapter contains some concluding remarks and open problems. 6 equiangular lines and spherical codes

2.1.1 Notation

We will always assume that the dimension n → ∞ and write f = o(g), respectively f = O(g) to mean f (n)/g(n) → 0 as n → ∞, respectively f (n)/g(n) ≤ C for some constant C and n sufficiently large. We will say γ is fixed to mean that it does not depend on n. n Let C = {v1,..., vm} be a spherical L-code in R . We define MC to be the associ- D E ated m × m Gram matrix given by (MC)i,j = vi, vj . We also define an associated complete edge-labelled graph GC as follows: let C be its vertex set and for any distinct u, v ∈ C, we give the edge uv the value γ iff hu, vi = γ. We also say that uv is a γ-edge and for brevity, we sometimes refer to γ as the “angle” between u and v, instead of the “cosine of the angle”. For β > 0, we slightly abuse our notation and say that uv is a β-edge if hu, vi ≤ −β. We call a subset S ⊂ GC a γ-clique if uv is a γ-edge for all distinct u, v ∈ S. For any x ∈ GC we define the γ-neighborhood of x to be Nγ(x) = {y ∈ GC : xy is a γ-edge}. Furthermore, we define the γ-degree dγ(x) = |Nγ(x)| and the maximum γ-degree ∆γ = maxx∈G dγ(x). We denote the identity matrix by I and denote the all 1’s matrix by J, where the size of the matrices is always clear from context. Let Y be a set of vectors in Rn. We define span(Y) to be the subspace spanned by the vectors of Y and for a subspace U, define U⊥ = {x ∈ Rn : hx, yi = 0 for all y ∈ U} to be the orthogonal n complement. For all x ∈ R define pY(x) to be the normalized (i.e. unit length) projection of x onto the orthogonal complement of span(Y), provided that the projection is nonzero. That is, if we write x = u + v for u ∈ span(Y)⊥, v ∈ span(Y) and u 6= 0, then pY(x) = u/kuk. More generally, for a set of vectors S we write pY(S) = {pY(x) : x ∈ S}.

2.2 equiangular lines

Suppose that we are given a set of equiangular lines in n-dimensional Euclidean space Rn with common angle arccos α. By identifying each line with a unit vector along this line, we obtain a set of unit vectors with the property that the inner product of any two vectors equals either α or −α. As we have already mentioned in the introduction, we will refer to such a set as a {−α, α}-code. Given a {−α, α}- code C, we call an α-edge of GC positive and a −α-edge negative. Van Lint and Seidel [66] observed that a particular set of equiangular lines corresponds to various {−α, α}-codes, depending on which of the two possible vectors we choose along each line. Conversely, this means that we can negate any number of vectors in a {−α, α}-code without changing the underlying set of equiangular lines. In the corresponding graph, this means that we can switch all the edges adjacent to some vertex from positive to negative and vice versa. The proof of Theorem 2.1.1 builds on several key observations. The first is that we can use Ramsey’s theorem to find a large positive clique in GC. We then negate some vertices outside of this clique, in order to obtain a particularly advantageous 2.2 equiangularlines 7 graph, for which we can show that almost all vertices attach to this positive clique entirely via positive edges. We then project this large set onto the orthogonal complement of the positive clique. Next we observe that the resulting graph contains few negative edges, which implies that the diagonal entries of the Gram matrix of the projected vectors are significantly larger in absolute value than all other entries. Combining this with an inequality which bounds the rank of such matrices already gives us a bound of (2 + o(1))n. To prove the exact result, we use more carefully the semidefiniteness of the Gram matrix together with some estimates on the largest eigenvalue of a graph. We finish this discussion by giving an example of an equiangular set of 2n − 1 n 2 lines with common angle arccos 3 in R , first given by Lemmens and Seidel 1 1 [64]. This is equivalent to constructing a spherical {− 3 , 3 }-code C of size 2n − 2. For any such code C, observe that the Gram matrix MC is a symmetric, positive- semidefinite (2n − 2) × (2n − 2) matrix with 1s on the diagonal and rank at most n. Conversely, if M is any matrix satisfying all properties listed, then M is the Gram matrix of a set of 2n − 2 unit vectors in Rn, see e.g. [42, Lemma 8.6.1]. Thus it suffices to construct such a matrix. To that end, consider the matrix M with n − 1 blocks on the diagonal, each of the form ! 1 − 1 3 , 1 − 3 1

1 and all other entries 3 . Clearly M is a (2n − 2) × (2n − 2) symmetric matrix, so we just have to verify that it is positive-semideﬁnite and of rank n. To do so, we need to show that M has smallest eigenvalue 0 with multiplicity n − 2. This is a routine calculation.

2.2.1 Orthogonal projections

Before we can delve into the proof of Theorem 2.1.1, we will set the ground by providing some necessary lemmas. We start with a well-known upper bound on the size of a negative clique, which will guarantee us a large positive clique using Ramsey’s theorem. For later purposes, the lemma is stated in some more generality. Lemma 2.2.1. Let 0 < α < 1 and let C be a spherical [−1, −α]-code in Rn. Then |C| ≤ α−1 + 1.

0 Proof. Let v = ∑x∈C x. Then, since every x ∈ C is a unit vector and hx, x i ≤ −α for x 6= x0, 0 ≤ kvk2 = ∑ kxk2 + ∑ x, x0 ≤ |C| − α|C|(|C| − 1), x∈C x,x0∈C x6=x0 which we can rewrite into the desired upper bound on |C|. 8 equiangular lines and spherical codes

Remark 2.2.2. We note that equality in the above lemma occurs only if the vectors of C form a regular simplex. As indicated above, this lemma enables us to ﬁnd a large positive clique in our graph. The next step is to understand how the remaining vertices attach to this clique. A key tool towards this goal is orthogonal projection. We will ﬁrst need a lemma that lets us compute the inner product between two vectors in the span of a clique in terms of the inner products between the vectors and the clique. Because we will need it again in a later section, we state it in some generality. Lemma 2.2.3. Let −1 ≤ γ < 1 and t 6= −1/γ + 1. Suppose Y is a spherical {γ}-code of size t and V is the matrix with Y as column vectors. Then for all v1, v2 ∈ span(Y) we have | − γ | s1 s2 1+γ(t−1) s1 Js2 hv , v i = , 1 2 1 − γ | where si = V vi for i = 1, 2 are the vectors of inner products between vi and Y.

Proof. Let us ﬁrst prove that Y is linearly independent. Suppose that ∑y∈Y cyy = 0 0 for some reals cy. Taking the inner product with some y ∈ Y gives 0 0 = ∑ cy y, y = (1 − γ)cy0 + γ ∑ cy. y∈Y y∈Y

0 Since this equation is true for all y ∈ Y and γ 6= 1, all cy are identical. Unless they equal 0, this implies that 1 + (t − 1)γ = 0, a contradiction. By passing to a subspace, we may assume that Y ⊂ Rt, so that V| is invertible | −1 and we have vi = (V ) si. Thus

| −1 | | −1 | −1 | −1 | | −1 hv1, v2i = ((V ) s1) (V ) s2 = s1 V (V ) s2 = s1 (V V) s2. To obtain the result, we observe that V|V is the Gram matrix of Y, so that V|V = (1 − γ)I + γJ and moreover − γ I 1+γ(t−1) J (V|V)−1 = . 1 − γ The following lemma shows how the angle between two vectors changes under an appropriate projection. Recall that pY(x) denotes the normalized projection of x onto the orthogonal complement of span(Y). n Lemma 2.2.4. Let −1 < γ < 1 and let Y ∪ {x1, x2} be a set of unit vectors in R so that all pairwise inner products, except possibly hx1, x2i, equal γ. Suppose additionally that Y has size 1 if γ is negative. Then pY(x1) and pY(x2) are well-deﬁned and we have hx , x i − γ γ(1 − hx , x i) hp (x ), p (x )i = 1 2 + 1 2 .(2.3) Y 1 Y 2 1 − γ (1 + γ|Y|)(1 − γ) 2.2 equiangularlines 9

⊥ Proof. For i = 1, 2, write xi = ui + vi where vi ∈ span(Y) and ui ∈ span(Y) . Let | | V be the matrix with Y as columns and observe that si = V vi = V (xi − ui) = (γ,..., γ)|. Let t = |Y| and observe that t 6= −1/γ + 1 and t 6= −1/γ, so we can apply Lemma 2.2.3 to obtain

γ 2 − ( )2 2 tγ 1+γ(t−1) tγ tγ hv , v i = hv , v i = hv , v i = = < 1. 1 2 1 1 2 2 1 − γ 1 + γ(t − 1)

Thus using the fact that xi is a unit vector and ui, vi are orthogonal, we have 2 2 ||ui|| = 1 − ||vi|| > 0 for i = 1, 2. Since pY(xi) = ui/||ui||, we can ﬁnish the proof by computing

tγ2 hu , u i hx , x i − hv , v i hx1, x2i − + ( − ) 1 2 = 1 2 1 2 = 1 γ t 1 p p tγ2 ku1kku2k 1 − ||v ||2 1 − ||v ||2 − 1 2 1 1+γ(t−1) hx , x i − γ γ(1 − hx , x i) = 1 2 + 1 2 . 1 − γ (1 + γt)(1 − γ) Remark 2.2.5. Note that when the conditions of the lemma are met we have hpY(x1), pY(x2)i ≤ hx1, x2i, which in particular implies that pY(x1) 6= pY(x2) when x1 6= x2. Note furthermore that if |Y| = 1, then the right-hand side of (2.3) simpli- 2 2 fies to (hx1, x2i − γ )/(1 − γ ) (this is most easily seen by looking at the second- to-last term in the final equation of the above proof) and that, for fixed hx1, x2i, the latter is a decreasing function in γ2. In particular, after projecting onto a positive clique (i.e. γ = α) of size t, an angle −1 0 0 −1 of α becomes 1/(t + α ) (that is, if hx, x i = α, then hpY(x), pY(x )i = 1/(t + α )) and an angle of −α becomes 2α 1 + α − + . 1 − α (t + α−1)(1 − α) Since these two angles will frequently pop up, we will make the following definition. Definition 2.2.6. For α ∈ (0, 1) and t ∈ N, let L(α, t) = {−σ(1 − e) + e, e}, where e = e(α, t) = 1/(t + α−1) and σ = σ(α) = 2α/(1 − α). Note that L(α, t) comprises the two possible angles after projecting onto a positive clique of size t. A set attached to a positive clique in a {−α, α}-code entirely via positive edges therefore turns into an L(α, t)-code after projecting. When we project, we will continue to call edges positive or negative according to whether their original values are α or −α. Note in particular that a positive edge may obtain a negative value after projection. Equipped with this machinery to handle projections, the next lemma gives an upper bound on the number of vertices which are not attached to the positive clique entirely via positive edges. The result is analogous to Lemma 5 of Bukh [16]. 10 equiangular lines and spherical codes

Lemma 2.2.7. Let X ∪ Y ∪ {z} be a {−α, α}-code in Rn in which all edges incident to any y ∈ Y are positive and all edges between X and z are negative. If |Y| ≥ 2/α2, then |X| < 2/α2.

Proof. Let us ﬁrst project X ∪ {z} onto the orthogonal complement of span(Y), and 0 0 let us denote pY(X) by X and pY(z) by z . By Lemma 2.2.4 and the subsequent paragraph, we verify that X0 ∪ {z0} is an L(α, |Y|)-code in which all edges incident to z0 are negative and have value −σ(1 − e) + e, which, by Remark 2.2.5, is at most −α. The positive angles equal e < 1/|Y| and since |Y| ≥ 2/α2 we get the bound e ≤ α2/2. Let us now project X0 onto the orthogonal complement of span(z0). By Lemma 2.2.4 and Remark 2.2.5 we ﬁnd that the positive angle becomes

e − (σ(1 − e) − e)2 (2.4) 1 − (σ(1 − e) − e)2

−α−(σ(1−e)−e)2 and the negative angles become at most 1−(σ(1−e)−e)2 , which is at most (2.4). Furthermore, using (σ(1 − e) − e)2 ≥ α2 and Remark 2.2.5,(2.4) is at most (e − α2)/(1 − α2). Since e ≤ α2/2, this yields an upper bound of −α2/(2 − 2α2) on all angles after projection. Therefore, after projecting X0 onto the orthogonal complement of span(z0), we obtain a spherical [−1, −α2/(2 − 2α2)]-code. By Lemma 2.2.1, it has size at most (2 − 2α2)/α2 + 1 < 2/α2, concluding the proof of the lemma.

Using this lemma, we will see that, after appropriately negating some vertices, all but a fixed number of vertices are attached to the positive clique via positive edges. Hence Theorem 2.1.1 can be reduced to studying L(α, t)-codes, as follows. Lemma 2.2.8. Let α ∈ (0, 1) be fixed and let t = log log n. For all sufficiently large n and for any spherical {−α, α}-code C in Rn, there exists a spherical L(α, t)-code C0 in Rn such that |C| ≤ |C0| + o(n).

Proof. Recall that GC denotes the complete edge-labelled graph corresponding to C. From Lemma 2.2.1 we know that GC doesn’t contain a negative clique of size α−1 + 2. By Ramsey’s theorem there exists some integer R such that every graph on at least R vertices contains either a negative clique of size at least α−1 + 2 or a positive clique of size t. A well-known bound of Erd˝osand Szekeres [34] shows R ≤ 4t = o(n). Thus if |C| < R, then we are done by taking C0 = ∅. Otherwise we have by Ramsey’s theorem that GC contains a positive clique Y of size t. For any T ⊂ Y, let ST comprise all vertices v in GC \ Y for which the edge vy (y ∈ Y) is positive precisely when y ∈ T. Let us negate all vertices v which lie in ST for some |T| < t/2 and note that C remains a {−α, α}-code. However, all sets ST for |T| < t/2 are now empty. Given some T ⊂ Y with t/2 ≤ |T| < t, pick a vertex z ∈ Y \ T and consider the {−α, α}-code ST ∪ T ∪ {z}. Since any edge incident to 2 T is positive, all edges between ST and z are negative and |T| ≥ t/2 > 2/α for n 2 large enough, we can apply Lemma 2.2.7 to deduce that |ST| < 2/α . Moreover, by 2.2 equiangular lines 11

0 Lemma 2.2.4 and Remark 2.2.5 we have that C = pY(SY) is an L(α, t)-code with 0 |C | = |SY|. Thus we conclude

0 t+1 2 0 |C| = |SY| + ∑ |ST| + |Y| < |C | + 2 /α + t = |C | + o(n). t/2<|T|

2.2.2 Spectral techniques

In view of Lemma 2.2.8, we just need to bound the size of L(α, t)-codes. Our main tool will be an inequality bounding the rank of a matrix in terms of its trace and the trace of its square. This inequality goes back to [11, p. 138] and its proof is based on a trick employed by Schnirelman in his work on Goldbach’s conjecture [80]. For various combinatorial applications of this inequality, see, for instance, the survey by Alon [3] and other recent results [31]. Lemma 2.2.9. Let M be a symmetric real matrix. Then rk(M) ≥ tr(M)2/tr(M2).

Proof. Let r denote the rank of M. Since M is a symmetric real matrix, M has r precisely r non-zero real eigenvalues λ1,..., λr. Note that tr(M) = ∑i=1 λi and 2 r 2 r 2 r 2 tr(M ) = ∑i=1 λi . Applying Cauchy–Schwarz yields r ∑i=1 λi ≥ (∑i=1 λi) , which is equivalent to the desired inequality.

We use Lemma 2.2.9 to deduce the next claim. Lemma 2.2.10. Let C be an L(α, t)-code in Rn and let d denote the average degree of the 2 graph spanned by the negative edges in GC. Then |C| ≤ (1 + σ d)(n + 1).

Proof. Recall that L(α, t) = {−σ(1 − e) + e, e}. Every diagonal entry of N = MC − eJ equals 1 − e and N contains exactly d|C| non-zero off-diagonal entries, each of which equals −σ(1 − e). Observe that rk(N) ≤ rk(MC) + rk(J) ≤ n + 1 by the 2 2 subadditivity of the rank. Furthermore, tr(N) = |C|(1 − e) and tr(N ) = ∑i,j Nij. By applying Lemma 2.2.9 to N we can therefore deduce |C|2(1 − e)2 ≤ |C|(1 − e)2 + |C|dσ2(1 − e)2 (n + 1), which is equivalent to the desired inequality after dividing by |C|(1 − e)2.

It thus proves necessary to obtain upper bounds on the average degree d of the negative edges in GC. Remark 2.2.11. The proof of Lemma 2.2.7 provides us with a bound on d, and if we are a bit more careful, we already have enough to prove that for a ﬁxed α and t → ∞, any L(α, t)-code has size at most 2n + o(n). Indeed, suppose that C is an L(α, t)-code with t → ∞. Let z0 ∈ C and let X0 be the vertices connected to z0 via negative edges. We project X0 onto the orthogonal complement of z0 and observe that since t → ∞, e → 0 and hence the positive angle (2.4) in Lemma 2.2.7 becomes −σ2/(1 − σ2) + o(1). Thus we obtain a [−1, −σ2/(1 − σ2) + o(1)]-code which has 12 equiangular lines and spherical codes

size at most (1 − σ2)/σ2 + 1 + o(1) = 1/σ2 + o(1) by Lemma 2.2.1. Since this holds for all z, we have that d ≤ 1/σ2 + o(1) and hence applying Lemma 2.2.10 we conclude |C| ≤ 2n + o(n). The following lemma shows that it will be sufficient to find an upper bound on d in terms of the largest eigenvalue of some fixed-size subgraph of C, by which we mean a subgraph of size O(1). Let us fix some standard notation. For a matrix A, we denote its largest eigenvalue by λ1(A). If H is a graph, then we can identify H with its adjacency matrix A(H), so that we will write λ1(H) to mean λ1(A(H)). It is well-known that λ1 is monotone in the following sense: if H is a subgraph of G, then λ1(H) ≤ λ1(G) (see e.g. [68, chapter 11, exercise 13]) Lemma 2.2.12. Let C be a fixed-size L(α, t)-code in Rn and assume that t → ∞ as n → ∞. Let H be the subgraph of GC containing precisely all negative edges. Then σλ1(H) ≤ 1 + o(1).

Proof. The Gram matrix MC and A = A(H) are related by the equation

MC = I + e(J − I) − σ(1 − e)A, where J denotes the all-ones matrix. Let x be a normalized eigenvector of A with eigenvalue λ1(H). Since MC is positive-semideﬁnite, we deduce

0 ≤ hMC x, xi = 1 − e + e hJx, xi − σ(1 − e)λ1(H) (2.5) ≤ 1 − σλ1(H) + e(|C| + σλ1(H)), where hJx, xi ≤ |C| follows from the fact that |C| is the largest eigenvalue of J. Since σ, |C| and λ1(H) are all O(1) and e = o(1),(2.5) yields the required σλ1(H) ≤ 1 + o(1).

The following two lemmas are concerned with establishing a connection between the average degree of a graph and its largest eigenvalue. The ﬁrst lemma and its proof are inspired by Nilli’s proof [75] of the Alon-Boppana bound on the second eigenvalue of a graph.

Lemma 2.2.13. Let G be a graph with minimum degree δ > 1. Let v0 be some vertex of G and let H be the subgraph√ consisting of all vertices within distance k of v0. Then λ1(H) ≥ 2(1 − 1/(k + 1)) δ − 1.

Proof. For 0 ≤ i ≤ k, let Vi denote the set of vertices at distance i from v0 in H, let ei denote the number of edges in H[Vi] and let hi denote the number of edges in H[Vi, Vi+1], where we set hk = 0 and, since we will need it later in the proof, −i/2 h−1 = 0. Let us deﬁne a function f on the vertices of H by f (v) = (δ − 1) if v ∈ Vi. Letting A denote the adjacency matrix of H, we have λ1(H) ≥ hA f , f i / h f , f i. In order to prove the desired bound on λ1(H), we therefore need to bound the quantity h f , f i from above in terms of hA f , f i. We have

k |V | k 2e 2h h f , f i = i and hA f , f i = i + i . ∑ i ∑ i i+1/2 i=0 (δ − 1) i=0 (δ − 1) (δ − 1) 2.2 equiangular lines 13

Note that for 0 ≤ i ≤ k − 1, hi−1 + 2ei + hi counts the sum of the degrees of all vertices in Vi and is therefore of size at least δ|Vi|. Moreover, since every vertex in Vi+1 is adjacent to some vertex in Vi we have |Vi+1| ≤ hi. Fix any j in the range 0 ≤ j ≤ k. Using the above two observations we ﬁnd

j−1 k−1 h + 2e + h |Vj| h h f f i ≤ i−1 i i + + i , ∑ i j ∑ i+1 .(2.6) i=0 δ(δ − 1) (δ − 1) i=j (δ − 1) √ Observe that δ ≥ 2 δ − 1 and that we have the identity 1 1 1 + = . δ(δ − 1)i δ(δ − 1)i+1 (δ − 1)i+1

Collecting terms belonging to the same hi in the ﬁrst sum of (2.6) and using the estimate and identity of the previous sentence, we ﬁnd

k h 2e |Vj| hA f , f i |Vj| h f f i ≤ i + i + ≤ √ + , ∑ i+1 i j j . i=0 (δ − 1) δ(δ − 1) (δ − 1) 2 δ − 1 (δ − 1) Averaging over all 0 ≤ j ≤ k yields

hA f , f i h f , f i h f , f i ≤ √ + , 2 δ − 1 k + 1 which is equivalent to the desired inequality.

Lemma 2.2.14. Let H be a connected graph on

(i) 11 vertices and 10 edges. Then λ1(H) ≥ 20/11.

(ii) k vertices and k edges. Then λ1(H) ≥ 2.

(iii) 6 vertices and 5 edges, so that some vertex has degree 5. Then λ1(H) ≥ 2.2.

(iv) 5 vertices and 5 edges so that some vertex has degree 4. Then λ1(H) ≥ 2.25.

(v) 8 edges so that some vertex has degree 4. Then λ1(H) ≥ 2.2.

Proof. Let A denote the adjacency matrix of H and 1 the all-ones vector of appro- 1 1 1 1 priate length. Note that λ1(H) ≥ hA , i / h , i = d, where d denotes the average degree of H. This is sufﬁcient to establish (i) and (ii), since the average degree of the graphs is 20/11 and 2, respectively. Suppose√ that H is a star with 5 leaves, as in (iii). Let x be the vector giving weight 5 to√ its internal vertex and weight 1 to√ each leaf. Then hx, xi = 10 and hAx, xi = 10 5 yielding the required λ1(H) ≥ 5 > 2.2. Suppose that H is as in (iv). Let x be the vector giving weight 1 to the vertex of degree 4 and 1/2 to the others. Then hx, xi = 2 and hAx, xi = 4.5 yielding the required λ1(H) ≥ 2.25. 14 equiangular lines and spherical codes

Finally, suppose that H is as in (v) and let v be the vertex of degree 4. If two of the neighbors of v are adjacent,√ we are done by (iv). Otherwise, let x be the vector giving weight 4 to√v, weight 5 to its 4 neighbors and weight 1 to all other vertices. Then hAx, xi = 40 5 and hx, xi ≤√40 since there are at most 4 vertices of weight 1. Hence λ1(H) ≥ hAx, xi / hx, xi ≥ 5 > 2.2.

The next lemma deals with {−α, α}-codes in which the negative edges are very sparse. This will be the case when α is rather large. 1 n Lemma 2.2.15. Let α ∈ (0, 1) \{ 3 } and let C be an L(α, t)-code in R . If the negative edges form a matching, then |C| ≤ n + 1.

Proof. Recall that L(α, t) = {−σ(1 − e) + e, e}. Let J denote the all-ones matrix. Since the rank of matrices is subadditive, we have

rk(MC − eJ) ≤ rk(MC) + rk(−eJ) = rk(MC) + 1 ≤ n + 1. (2.7)

Since the negative edges of GC form a matching, the matrix (MC − eJ)/(1 − e) consists of m identical 2 × 2 blocks with 1’s on the diagonal and −σ off the diagonal, and |C| − 2m identical 1 × 1 identity matrices, where m denotes the number of 2 1 negative edges. The former have determinant 1 − σ , the latter 1. Since α 6= 3 , these quantities are non-zero, so that MC − eJ has full rank, that is, rk(MC − αJ) = |C|. Together with (2.7) this gives the desired inequality.

Remark 2.2.16. Note that one can also prove |C| ≤ n with some more work.

2.2.3 Proof of the main result

In this section, we present the proof of Theorem 2.1.1. First, combining Lemma 2.2.10 with the newly gained information about the relation between σ, the largest eigenvalue of ﬁxed-size graphs and d, we prove the following theorem about L(α, t)- codes. This theorem will allow us to analyse equiangular lines for all angles except 1 arccos 3 . 1 Theorem 2.2.17. Let α ∈ (0, 1) \{ 3 } and t ∈ N so that t → ∞ as n → ∞. Let C be an L(α, t)-code in Rn for which every vertex is incident to at most O(1) negative edges. Then |C| < 1.92n for sufﬁciently large n.

Proof. Recall that L(α, t) = {−σ(1 − e) + e, e}, where e = 1/(t + α−1) and σ = 2α/(1 − α); note that e = o(1). Throughout the proof, let G denote the graph consisting only of the negative edges of the graph corresponding to C (that is, we delete from GC all positive edges to obtain G). We split the proof of this lemma into different regimes, depending on the value of σ. Case 1, σ ∈ [0.71, ∞): We will show that no two edges in G are adjacent. Together with Lemma 2.2.15 this will show that |C| ≤ n + 1. Let β = −σ(1 − e) + e. If β < −1, G cannot contain any edges and we are done. Otherwise, suppose to the contrary 2.2 equiangular lines 15 that x, y and z are unit vectors in C so that xy and xz are negative edges. Let us decompose y and z as y = βx + u and z = βx + v, where u and v are orthogonal to x. Since x, y and z are unit vectors, taking norms on both sides of√ each equation and 2 2 2 rearranging yields 1 − β = kuk = kvk . Since σ ≥ 0.71 > 1/ 2 and e = o(1),√ we have β2 > 1/2 + e for sufficiently large n and hence t. Therefore, kuk, kvk < 1/ 2. Furthermore, taking the inner product of y and z gives hy, zi = β2 + hu, vi > e + 1/2 − kukkvk > e, a contradiction to hy, zi ∈ L(α, t), finishing the proof of the first case. Case 2, σ ∈ [0.551, 0.71]: We will prove that G decomposes into trees on at most 10 vertices. Lemma 2.2.12 shows that G cannot contain a fixed-size subgraph H with λ1(H) > 1/0.55 = 20/11. In particular, by Lemma 2.2.14, G doesn’t contain a subgraph on 11 vertices and 10 edges or a subgraph on k vertices and k edges for any k ≤ 10. Since any connected graph on at least 11 vertices contains a tree on 11 vertices, all components have at most 10 vertices, and since the only acyclic components are trees, all components are trees on at most 10 vertices. The average degree of any component is therefore at most 18/10 and hence so is the average degree of G. Applying Lemma 2.2.10 establishes the required bound |C| ≤ (1 + 1.8σ2)(n + 1) < 1.92n. Case 3, σ ∈ [0.47, 0.551]: Lemma 2.2.12 implies that G cannot contain a fixed- size subgraph H with λ1(H) > 2.13 > 1/0.47. We can therefore deduce from Lemma 2.2.14 that G doesn’t contain a vertex of degree higher than 4, that the neighborhood of a vertex of degree 4 contains no edges and that the neighborhood of a vertex of degree 4 is incident to at most 3 more edges. The latter two properties imply that each vertex of degree 4 is adjacent to a leaf. On the other hand, each leaf is adjacent to exactly one vertex (not necessarily of degree 4), so G contains no more vertices of degree 4 than leaves. Since G also doesn’t contain any vertices of higher degree than 4, the average degree of G is at most 3. Applying Lemma 2.2.10 establishes the required bound |C| ≤ (1 + 3σ2)(n + 1) < 1.92n.

Case 4, σ ∈ (0, 0.47]: Let d be the average degree of the negative edges in GC and suppose for the sake of contradiction that |C| > 1.92n. Combining this lower bound on |C| with the upper bound given by Lemma 2.2.10 yields d > 0.92/σ2 − o(1) > 4. Let l be the integer satisfying 2l < d ≤ 2l + 2; note that d > 4 implies l ≥ 2. It is well known that a graph with average degree d contains a subgraph with minimum degree at least d/2. Hence G contains a subgraph G0 with minimum degree at least 0 0 l + 1. Applying Lemma 2.2.13 to G for k = √11, we ﬁnd that G contains a subgraph H with maximal eigenvalue λ1(H) > 1.83 l and, since the maximum degree of G is bounded by a constant independent of n, so is the size of H by construction. Lemma 2.2.12 then gives 1 + o(1) 1 σ2 ≤ < , 1.832l 3.34l 16 equiangular lines and spherical codes

which together with Lemma 2.2.10 and l ≥ 2 yields the required

2(l + 1) |C| < 1 + (n + 1) < 1.92n. 3.34l

Now that we have ﬁnished all the necessary preparation, we are ready to complete the proof of our ﬁrst theorem.

Proof of Theorem 2.1.1. Let C be a {−α, α}-code in Rn and let t = log log n. Sup- 1 0 n pose first that α 6= 3 . Then by Lemma 2.2.8, there exists an L(α, t)-code C in R such that |C| ≤ |C0| + o(n). By Theorem 2.2.17, we have |C0| < 1.92n and hence |C| ≤ 1.93n for n large enough. 1 Otherwise α = 3 . For a detailed proof of the upper bound of 2n − 2, we refer the reader to [64]. Let us nonetheless sketch it for the sake of completeness. Note that what follows is only an outline; filling in all the details requires substantially more work. Instead of finding a large positive clique, we consider the largest negative clique M in any graph obtained from GC by switching any number of vertices. By Lemma 2.2.1, we have that |M| ≤ 4. We can then show that the cases |M| ≤ 3 are either straightforward or can be reduced to the case |M| = 4. In the latter case, we can show that unless all vertices attach to M in the same way (that is, no two vertices outside the clique attach to some vertex within the clique differently), |C| is bounded from above by some constant independent of n. If they do all attach 0 in the same way, then if we consider the projection C = pM(C\M) of C\M onto the orthogonal complement of span(M), we obtain a {−1, 0}-code. This means that any two distinct vectors of C0 are either orthogonal or lie in the same 1-dimensional subspace, so that |C0| ≤ 2dim(C0). Moreover, by Remark 2.2.2 M is a regular simplex so it lives in a 3-dimensional subspace, and hence dim(C0) = n − 3. Thus |C| = |M| + |C0| ≤ 4 + 2(n − 3) = 2n − 2, finishing the proof.

2.3 spherical codes

Let us now turn our attention from equiangular sets of lines to the more general setting of spherical codes. Recall that a spherical L-code is a finite non-empty set C of unit vectors in Euclidean space Rn so that hx, yi ∈ L for any pair of distinct vectors x, y in C. In this section, we prove Theorem 2.1.4 in the case k = 1, obtaining the asymptotically tight bound even when α is allowed to depend on n. The proof features all ideas central to the argument in the multi-angular case (which we will treat in detail in Section 2.5), without concealing them unnecessarily. Theorem 2.3.1. Let β ∈ (0, 1] be fixed and α ∈ [−1, 1). Then any [−1, −β] ∪ {α}-code in Rn has size at most α 2 1 + max , 0 n + o(n). β Since an equiangular set of lines corresponds to a {−α, α}-code, this implies a weaker bound of 4n + o(n) for equiangular sets. The reason for this is that we can’t 2.3 spherical codes 17 switch edges from negative to positive any more, since a negative edge might not obtain value α after switching. Moreover, this is essentially tight because if we take 1 our example of 2n − 2 lines with angle arccos 3 and take both unit vectors along 1 1 each line, we get a [−1, − 3 ] ∪ { 3 }-code of size 4n − 4. The beginning of the proof of Theorem 2.3.1 is along the lines of the proof of the corresponding theorem for equiangular sets of lines. We start by finding a large positive clique in GC. Unlike before, however, a substantial portion of the vertices might not attach to this clique entirely via positive edges. In fact, almost all vertices attach either entirely via positive edges or mostly via negative ones. Similarly to before, we can bound the size of the set of vertices attaching positively to the clique by 2n + o(n). Repeating this argument yields a set of positive cliques in such a way that almost all edges between these cliques are negative. This imposes a bound on the number of repetitions, which is enough to bound the size of the L-code. We start by proving a lemma similar to Lemma 2.2.7, which enables us to analyse how vertices connect to a positive clique. Lemma 2.3.2. Let L = [−1, −β] ∪ {α} for some α, β ∈ (0, 1) and suppose that X ∪ Y ∪ {z} is an L-code in which all edges incident to any y ∈ Y are positive edges and all edges between X and z are negative edges. Suppose furthermore that |Y| > 1/α2. Then |X| < 1/β2.

Proof. Let αX denote the average value of the edges in X and −βz the average value of the edges between X and z. Note that αX ≤ α and βz ≥ β. Let M be the Gram matrix of X ∪ Y ∪ {z} and let v = (x,..., x, y,..., y, ζ)|, where

α(1 + βz)/|Y| x = 1/|X|, y = − and ζ = βz − y|Y|α. α − α2 + (1 − α)/|Y| Then hMv, vi 2 (|X| − |X|)αX + |X| 2 2 2 = + 2y|Y|α + y ((|Y| − |Y|)α + |Y|) + 2ζ(−βz + y|Y|α) + ζ |X|2 1 − α = X + α − β2 + 2y|Y|α(1 + β ) + y2|Y|2α − α2 + (1 − α)/|Y| |X| X z z 1 − α α2(1 + β )2 = X + α − β2 − z |X| X z α − α2 + (1 − α)/|Y| 1 − α α2(1 + β )2 ≤ + α − β2 − z . |X| z α − α2 + (1 − α)/|Y|

2 2 Since hMv, vi ≥ 0 and βz ≥ β , it is therefore sufﬁcient to prove that

α2(1 + β )2 α − β2 − z < −β2(1 − α). z α − α2 + (1 − α)/|Y| z 18 equiangular lines and spherical codes

Using |Y| > 1/α2 and rewriting the above inequality, it sufﬁces to show that

α(1 + β )2 α(1 − β2) < z , z 1 − α2

which is clearly true since α, βz > 0.

Remark 2.3.3. The v in the above proof is chosen so as to minimize hMv, vi /||v||2. An appropriate projection also minimizes this quantity and so the above argument could also be done using projections. Indeed, this minimization is precisely why projections are so useful for us. After projecting onto a large α-clique, the new α will become o(1). In this case, the next lemma gives a bound on the values of the negative edges incident to a ﬁxed vertex. Lemma 2.3.4. Let L = [−1, −β] ∪ {α} and let C be an L-code. If α = o(1) and −β1,..., −βN are the values of the negative edges incident to some vertex x in GC, then N 2 αN = o(1) and ∑i=1 βi ≤ 1 + o(1).

Proof. We will ﬁrst derive the upper bound on N. Let C = Nβ(x) ∪ {x} and M = | MC. If we let v = (1, . . . , 1, βN) , then we have

N 2 2 2 2 2 0 ≤ hMv, vi = ∑ Mij − 2βN ∑ βi + β N ≤ N + o(1)N − β N , 1≤i,j≤N i=1

which implies N ≤ (1 + o(1))β−2 and therefore establishes the claimed αN ≤ −2 | (1 + o(1))αβ = o(1). Now if we let w = (β1,..., βN, 1) , then we obtain

N N 2 2 hMw, wi = 1 − ∑ βi + ∑ βi βj Mij ≤ 1 − ∑ βi + ∑ βi βjα i=1 1≤i,j≤N i=1 1≤i,j≤N i6=j N N 2 2 ≤ 1 − ∑ βi + αN ∑ βi , i=1 i=1 where the last step follows from Cauchy–Schwarz. Using hMw, wi ≥ 0 and αN = N 2 o(1), we obtain the required ∑i=1 βi ≤ 1 + o(1). As we outlined above, when proving Theorem 2.3.1 we will obtain a multipartite graph which has mostly negative edges between its parts. The next lemma gives a bound on the number of parts of such a graph. Because we will consider more general spherical codes in a later section, we prove it in more generality. Lemma 2.3.5. Let β ∈ (0, 1] be ﬁxed, let α ∈ [−1, 1) and L = [−1, −β] ∪ [α, 1). Suppose t → ∞ as n → ∞ and let C be a spherical L-code such that GC is the disjoint union of ` α-cliques Y1,..., Y` each of size t, such that the number of β-edges between any Yi and Yj is at least t2(1 − o(1)). Then ` ≤ 1 + α/β + o(1). 2.3 spherical codes 19

Proof. Let A be the number of α-edges and B be the number of β-edges in GC. |C| − − Since the remaining ( 2 ) A B edges have value at most 1, we have as in the proof of Lemma 2.2.1 that

0 ≤ ∑ x ≤ |C| − 2Bβ + 2Aα + |C|(|C| − 1) − 2B − 2A, x∈C

2 t which implies 2B(β + 1) + 2A(1 − α) ≤ |C| . Now observe that C has size `t, `(2) ` 2 α-edges inside the parts, and at least (2)t (1 − o(1)) β-edges between parts. Thus if we substitute these values into the above inequality and solve for `, we obtain the required β + α + 1−α − o(1)(1 + β) α ` ≤ t = 1 + + o(1). β − o(1)(1 + β) β Now we have all of the necessary tools to prove the main theorem of this section.

Proof of Theorem 2.3.1. Suppose ﬁrst that |α| < 1/ log log n = o(1). Let Q = MC − αJ. By the subadditivity of the rank we have rk(Q) ≤ rk(MC) + rk(J) ≤ n + 1. Consider some x ∈ C. Let N = dβ(x) and let −β1,..., −βN be the values of the N 2 negative edges incident to x. By Lemma 2.3.4 we have ∑j=1 βj ≤ 1 + o(1) and αN = o(1). It follows that if i is the row corresponding to x in N, then

N N 2 2 2 2 ∑ Qi,j = ∑ (βj + α) ≤ ∑ βj + 2|α|N + α N ≤ 1 + o(1). j6=i j=1 j=1

Noting that Q has 1 − α on the diagonal, we obtain

|C| |C| 2 2 2 tr(Q ) = ∑ Qi,i + ∑ ∑ Qi,j ≤ |C|(1 − α) + |C| (1 + o(1))) ≤ |C|(2 + o(1)). i=1 i=1 j6=i

Thus applying Lemma 2.2.9 to Q yields

|C|2(1 − α)2 = tr(Q)2 ≤ tr(Q2)rk(Q) ≤ |C|(2 + o(1))(n + 1).

After dividing by |C|(1 − α)2 = |C|(1 − o(1)), we obtain the required |C| ≤ 2n + o(n). We now prove the theorem for all remaining values of α, that is, for all α satisfying |α| ≥ 1/ log log n. If α < 0 we are done by Lemma 2.2.1. Suppose therefore that 1 α > 0. Let ` = 1 + α/β and t = 4 log n. Suppose for the sake of contradiction that there exists some e > 0 so that for arbitrarily large n,

|C| > 2`(1 + 2e)n.

From Lemma 2.2.1 we know that GC doesn’t contain a negative clique bigger than β−1 + 2. By Ramsey’s theorem, there exists some integer R so that every graph on at least R vertices contains either a negative clique of size at least β−1 + 2 or 20 equiangular lines and spherical codes

a positive√ clique of size t. A well-known bound of Erd˝osand Szekeres [34] shows t R ≤ 4 ≤ n < |GC|. Therefore GC contains a positive clique Y of size t. For any T ⊂ Y, let ST comprise all vertices v in GC \ Y for√ which vy (y ∈ Y) is an α-edge precisely when y ∈ T. Given some T ⊂ Y with t ≤ |T| < t, pick a vertex z ∈ Y \ T and consider the [−1, −β] ∪ {α}-code ST ∪ T ∪ {z}. Since any edge√ incident to T is an α-edge, all edges between ST and z are β-edges and |T| ≥ 2 2 t > 1/α , we can apply Lemma 2.3.2 to deduce that |ST| < 1/β . For T = Y, 0 0 since pY(SY) is a [−1, −β] ∪ {α }-code for α = 1/(t + 1/α) < 1/ log log n, we infer |SY| < (2 + e)n for sufﬁciently large n from the ﬁrst part of the proof. Now let 0 [ G = GC \ Y ∪ ST T⊂Y√ |T|> t

0 and note that for all x ∈ G , |Nα(x) ∩ Y| = o(t). Applying the bounds derived above, we obtain

[ Y ∪ S ≤ t + 2t/β2 + |S | ≤ 2(1 + e)n. T Y T⊂Y√ |T|> t

We can therefore iterate this procedure ` times to obtain ` disjoint α-cliques Y1,..., Y` 0 √ and a disjoint graph G of size at least 2en > n, so that the√ number of α-edges 2 0 between Yi and Yj is o(t ) for distinct i and j. Since |G | > n, there exists an 0 2 additional α-clique Y`+1 ⊂ G of size t, also with o(t ) edges to any Yi. But then the induced subgraph on Y1 ∪ ... ∪ Y`+1 contradicts Lemma 2.3.5, ﬁnishing the proof.

2.4 a construction

In this section we prove Theorem 2.1.5, which states that√ for any positive integers n, k, r and α1 ∈ (0, 1) with k and α1 fixed and r ≤ n, there exist α2,..., αk, p n n+r β = α1/r − O( log(n)/n) and a spherical L-code of size (1 + r)(k) in R with L = [−1, −β] ∪ {α1,..., αk}. This construction shows that the second statement of Theorem 2.1.4 is tight up to a constant factor. It also answers a question of Bukh. In [16] he asked whether for fixed α, any spherical [−1, 0) ∪ {α}-code has size at most linear in the dimension. Theorem 2.1.5 gives an example of such a code with size that√ is superlinear in the dimension. Indeed, for any α fixed, if we choose r = d n/ log(n)e then by Theorem 2.1.5, we obtain a [−1, −β] ∪ {α}-code of size at least rn ≥ n3/2/ log n in R(1+o(1))n, where β > 0 for n large enough. Given vectors u ∈ Rn1 and v ∈ Rn2 , we let (u, v) denote the concatenated vector in Rn1+n2 . We first give an outline of the construction. We start by finding a n {0, 1/k,..., (k − 1)/k}-code C of size (k), given by Lemma 2.4.1. We then take a regular r-simplex so that all inner products are negative. For each vector v of the simplex, we take a randomly rotated copy Cv of C and attach a scaled Cv to v 2.4 a construction 21 by concatenation, and then√ normalize all vectors to be unit length. That is, for all 2 u ∈ Cv we take (λu, v)/ λ + 1 where λ is a scaling factor chosen so that the resulting code has the given α1 as one of its inner products. By randomly rotating the copies of C, we ensure that the inner products between vectors coming from different copies remain negative. This follows from the well-known fact that the inner√ product between random unit vectors is unlikely to be much bigger than 1/ n, given by Lemma 2.4.2. Lemma 2.4.1. For any positive integers n, k with k ≤ n, there exists a spherical n n {0, 1/k,..., (k − 1)/k}-code of size (k) in R . n n Proof. Let C be the set of {0, 1}-vectors in R having exactly k 1’s. Then |C| = (k) 2 and for any distinct u, v ∈ C, we observe√ that hu, vi ∈ {0, 1, . . . , k − 1}. Since kuk = k for all u ∈ C, we thus obtain that C/ k is a {0, 1/k,..., (k − 1)/k}-code.

The following lemma follows from the well known bound for the area of a spherical cap, which can be found in [71, Corollary 2.2]. Lemma 2.4.2. Let u, u0 ∈ Rn be unit vectors chosen independently and uniformly at random. Then for all t > 0,

2 Pr [ u, u0 ≥ t] < e−t n/2.

We are now ready to prove Theorem 2.1.5.

Proof of Theorem 2.1.5. Let L = {0, 1/k,..., (k − 1)/k} and let C be an L-code of n √ size (k), as given by Lemma 2.4.1. Let λ = 1/α1 − 1 and deﬁne λ2(i − 1)/k + 1 α = i λ2 + 1 for 2 ≤ i ≤ k. Note that by the choice of λ, the above also holds for α1. Let 0 L = {α1,..., αk}. Let S be a set of r + 1 unit vectors in Rr so that hv, v0i = −1/r for all distinct 0 v, v ∈ S, i.e. S is a regular r-simplex. For each v ∈ S, let Cv be an independent and uniformly random rotation of C in Rn. We deﬁne 0 (λu, v) C = √ : v ∈ S, u ∈ Cv , λ2 + 1 and observe that k(λu, v)k2 = λ2 + 1, so that C0 is indeed a set of unit vectors in n+r n 0 R of size (1 + r)(k). Moreover, for any v ∈ S and distinct u, u ∈ Cv, we have (λu, v) (λu0, v) λ2 hu, u0i + 1 √ , √ = ∈ L0, λ2 + 1 λ2 + 1 λ2 + 1 0 0 0 since hu, u i ∈ L. Finally, suppose that u ∈ Cv and u ∈ Cv0 for distinct v, v ∈ S. Observe that u, u0 are independent and uniformly random unit vectors in Rn, so we 22 equiangular lines and spherical codes

q n 0 may apply Lemma 2.4.2 with t = 4 log (k) + 2 log n /n to obtain Pr [hu, u i ≥ ] < −t2n/2 = −1(n)−2 = 1/r−λ2t = − (p ( ) ) t e n k . Now deﬁne β λ2+1 α1/r O log n /n and observe that if hu, u0i < t, then

(λu, v) (λu0, v0) λ2 hu, u0i + hv, v0i λ2t − 1/r √ , √ = ≤ = −β. λ2 + 1 λ2 + 1 λ2 + 1 λ2 + 1 Thus it sufﬁces to show that with positive probability, hu, u0i < t for all possible u, u0, since then C0 will be a [−1, −β] ∪ L0-code. To that end, we observe that there |S| | |2 ≤ n 2 0 are ( 2 ) C n(k) such pairs u, u and so the result follows via a union bound.

2.5 lines with many angles and related spherical codes

2.5.1 A general bound when β is ﬁxed

In this section we give a proof of Conjecture 2.1.3, i.e. the ﬁrst statement of Theo- rem 2.1.4. To this end, we need a well-known variant of Ramsey’s theorem, whose short proof we include for the convenience of the reader. Let Kn denote the complete graph on n vertices. Given an edge-coloring of Kn, we call an ordered pair (X, Y) of disjoint subsets of vertices monochromatic if all edges in X ∪ Y incident to a vertex in Y have the same color. For the graph of a spherical code, we analogously call (X, Y) a monochromatic γ-pair if all edges incident to a vertex in Y have value γ. kt Lemma 2.5.1. Let k, t, m, n be positive integers satisfying n > k m and let f : E(Kn) → [k] be an edge k-coloring of Kn. Then there is a monochromatic pair (X, Y) such that |X| = m and |Y| = t.

Proof. Construct kt vertices v1,..., vkt and sets X1,..., Xkt as follows. Fix v1 arbitrarily and let c(1) ∈ [k] be a majority color among the edges (v1, u). Set X1 = kt−1 {u : f (v1, u) = c(1)}. By the pigeonhole principle, |X1| ≥ d(n − 1)/ke ≥ k m. In general, we ﬁx any vi+1 in Xi, let c(i + 1) ∈ [k] be a majority color among the edges (vi+1, u) with u ∈ Xi, and let Xi+1 = {u ∈ Xi : f (vi+1, u) = c(i + 1)}. Then kt−i−1 |Xi+1| ≥ d(|Xi| − 1)/ke ≥ k m, and for every 1 ≤ j ≤ i the edges from vj to all vertices in Xi+1 have color c(j). Since we have only k colors, there is a color c ∈ [k] and S ⊂ [kt] with |S| = t so that c(j) = c for all j ∈ S. Then Y = {vj : j ∈ S} and X = Xkt form a monochromatic pair of color c, satisfying the assertion of the lemma.

We will also need the following simple corollary of Turán’s theorem, which can be obtained by greedily deleting vertices together with their neighborhoods. Lemma 2.5.2. Every graph on n vertices with maximum degree ∆ contains an independent n set of size at least ∆+1 . 2.5 lines with many angles and related spherical codes 23

Finally, we will need the bound on the size of an L-code previously mentioned in the introduction, see [26, 60]. Lemma 2.5.3. If L ⊆ R with |L| = k and C is an L-spherical code in Rn then |C| ≤ n+k ( k ). Now we have the tools necessary to verify Conjecture 2.1.3.

Proof of ﬁrst part of Theorem 2.1.4. We argue by induction on k. The base case −1 is k = 0, when L = [−1, −β], and we can take cβ,0 = β + 1 by Lemma 2.2.1. 2kβ−1 Henceforth we suppose k > 0. We can assume n ≥ n0 = (2k) . Indeed, if we can prove the theorem under this assumption, then for n < n0 we can use the upper bound for Rn0 (since it contains Rn). Then we can deduce the bound for the k general case by multiplying cβ,k (obtained for the case n ≥ n0) by a factor n0 = 2k2 β−1 n (2k) . Now suppose C is an L-code in R , where L = [−1, −β] ∪ {α1,..., αk}, with α1 < ··· < αk. 2 −2 Consider the case αk < β /2. We claim that ∆β ≤ 2β + 1. Indeed, for any y, x1, x2 ∈ GC with hy, x1i , hy, x2i ≤ −β, we have by the proof of Lemma 2.2.4 that 2 hx1, x2i − hy, x1i hy, x2i αk − β py(x1), py(x2) = q q ≤ q q 2 2 2 2 1 − hy, x1i 1 − hy, x2i 1 − hy, x1i 1 − hy, x2i < −β2/2 .

−2 Thus the projection of the β-neighborhood of y satisﬁes |py(Nβ(y))| ≤ 2β + 1 by Lemma 2.2.1, as claimed. By Lemma 2.5.2, the graph of β-edges in GC has an −2 independent set S of size |C|/(2β + 2). Therefore S is an {α1,..., αk}-code, so k k −2 |S| ≤ n + 1 ≤ 2n by Lemma 2.5.3. Choosing cβ,k > 4β + 4, we see that the 2 theorem holds in this case. Henceforth we suppose αk ≥ β /2. 2 Next consider the case that there is ` ≥ 2 such that α`−1 < α` /2. Choosing the maximum such ` we have 2 2 4 2k−`+1 0 2k α` /2 = 2(α`/2) ≥ 2(α`+1/2) ≥ ... ≥ 2(αk/2) ≥ β := (β/2) .(2.8)

Note that by induction the graph of {β, α1,..., α`−1}-edges in GC contains no clique `−1 of order cβ,`−1n , so by Lemma 2.5.2 its complement has maximum degree at `−1 least m = |C|/(2cβ,`−1n ). Letting y ∈ GC be a vertex attaining this maximum degree in {α`,..., αk}-edges, we have by the pigeonhole principle that there exists J ⊂ GC of size at least m/k, and an index ` ≤ s ≤ k such that hx, yi = αs for all x ∈ J. Now observe that for any x1, x2 ∈ GC with hx1, x2i ∈ [−1, −β] ∪ {α1,..., α`−1}, we have by Lemma 2.2.4 and Remark 2.2.5 that hx , x i − α2 ( ) ( ) = 1 2 s ≤ 2 − 2 < − 2 ≤ − 0 py x1 , py x2 2 α` /2 α` α` /2 β . 1 − αs 0 Furthermore, by Lemma 2.2.4 we have that py(J) is an L -code, where 2 0 0 0 0 0 αi −αs L = [−1, −β ] ∪ {α ,..., α }, with α = 2 for i ≥ `. By the induction hypothesis, ` k i 1−αs 24 equiangular lines and spherical codes

k−`+1 we have |J| ≤ cβ0,k−`+1n , so choosing cβ,k > 2kcβ,`−1cβ0,k−`+1 the theorem holds in this case. 2 Now suppose that there is no ` > 1 such that α`−1 < α` /2. We must have 0 α1 > 0. Let t = d1/β e. We apply Lemma 2.5.1 to ﬁnd a monochromatic pair (X, Y) −(k+1)t with |Y| = t and |X| = m ≥ (k + 1) n. Since GC has no β-clique of size t by Lemma 2.2.1, (X, Y) must be a monochromatic αr-pair for some 1 ≤ r ≤ k. 0 Let X = pY(X) be the projection of X onto the orthogonal complement of Y. By 0 0 0 Lemma 2.2.4 and Remark 2.2.5, we have that X is a [−1, β] ∪ {α1,..., αk}-code, where 0 αi − αr αr(1 − αi) αi = + for 1 ≤ i ≤ k. 1 − αr (1 + αrt)(1 − αr) 0 2 (k+1)t −2 We can assume αk ≥ β /2, otherwise choosing cβ,k > (k + 1) (4β + 4) we 0 −1 −1 0 are done by the ﬁrst case considered above. Since αr = (αr + t) < β , the com- 2 putation in (2.8) implies that there exists ` > 1 such that α`−1 < α` /2. Choosing (k+1)t cβ,k > (k + 1) 2kcβ,`−1cβ0,k−`+1 we are done by the second case considered above.

2.5.2 An asymptotically tight bound when β, α1,..., αk are ﬁxed

The goal of this section will be to prove the second statement of Theorem 2.1.4. The case k = 1 is given by Theorem 2.3.1, so henceforth we assume that we are given a ﬁxed k ≥ 2. Our general strategy will be to use projections in order to reduce the number of positive angles and then apply induction. When projecting onto the orthogonal complement of a large clique, Lemma 2.2.4 tells us that the new inner product will be some function of the old one plus o(1). In view of this, it will be convenient to prove the following, slightly more general version of Theorem 2.1.4.

Theorem 2.5.4. Let β ∈ (0, 1], α1,..., αk ∈ [0, 1) be fixed with α1 < ... < αk and let n n ∈ N. If C is a spherical [−1, −β + o(1)] ∪ {α1 + o(1),..., αk + o(1)}-code in R , then α |C| ≤ 1 + 1 (k − 1)!(2n)k + o(nk) β for n sufficiently large. Remark 2.5.5. We use γ + o(1) to refer to a specific γ∗ ∈ R depending on n such that γ∗ = γ + o(1), not a range of possible values near γ. We will still say γ-edge, ∆γ, etc. when we are referring to a (γ + o(1))-edge, ∆γ+o(1), etc. in the graph GC. The case k = 1 of Theorem 2.5.4 follows by inserting o(1) terms into expressions in the proof of Theorem 2.3.1, and so we henceforth assume that it holds. Moreover, since we will be making use of induction, we also assume that Theorem 2.5.4 holds 0 for all k < k. Now let β ∈ (0, 1] and α1,..., αk ∈ [0, 1) be fixed with α1 < ... < αk and let n ∈ N. Let C be a spherical [−1, −β + o(1)] ∪ {α1 + o(1),..., αk + o(1)}-code in Rn. 2.5 lines with many angles and related spherical codes 25

The argument will be a generalization of the one used to prove Theorem 2.3.1 and so we will need to generalize some lemmas. Firstly, we will need the following generalization of Lemma 2.3.2. Lemma 2.5.6. Let α ∈ (0, 1) and γ ∈ [−1, 1) be distinct reals. Let X ∪ Y ∪ {z} be a set of unit vectors in Rn so that Y ∪ {z} is an {α}-code, that all edges inside X have value at most α, that all edges between X and Y have value at least α and that all edges between X and z all have value at least γ if γ > α and at most γ if γ < α. Suppose furthermore that |Y| > 4/α(γ − α)2. Then |X| < 1/(γ − α)2.

Proof. Let αX denote the average value of the edges in side X, αY the average value of the edges between X and Y and γz the average value of the edges between X and z. Note that our assumptions imply αY ≥ α ≥ αX and |γz − α| ≥ |γ − α|. Let M be the Gram matrix of X ∪ Y ∪ {z} and let v = (x,..., x, y,..., y, ζ)|, where

(αY − αγz)/|Y| x = 1/|X|, y = − and ζ = −(γz + y|Y|α). α − α2 + (1 − α)/|Y| Then, similarly to the proof of Lemma 2.3.2,

hMv, vi 2 |X| + (|X| − |X|)αX 2 2 2 = + 2y|Y|α + y ((|Y| − |Y|)α + |Y|) + 2ζ(γz + y|Y|α) + ζ |X|2 Y 1 − α = X + α − γ2 + 2y|Y|(α − αγ ) + y2|Y|2α − α2 + (1 − α)/|Y| |X| X z Y z 1 − α (α − αγ )2 ≤ + α − γ2 − Y z , |X| z α − α2 + (1 − α)/|Y|

2 where the last inequality uses αX ≤ α. Note that αY ≥ α ≥ αγz, so (αY − αγz) ≥ 2 2 2 2 α (1 − γz) . Since hMv, vi ≥ 0 and (γz − α) ≥ (γ − α) , it is therefore sufﬁcient to prove that 2 2 2 α (1 − γz) 2 α − γ − < −(γz − α) . z α − α2 + (1 − α)/|Y|

As one can easily check, this can be rewritten as |Y| ≥ (1 − α)(1 + α − 2γz)/α(γz − α)2, which is true by assumption.

We will also need the following generalization of Lemma 2.3.4.

Lemma 2.5.7. If α1 = 0, then GC has the following degree bounds: (i) ∆ ≤ 1 (k − 2)!(2n)k−1 + o(nk−1) for all 2 ≤ i ≤ k. αi αi

(ii) If −β1,..., −βN are the values of the β-edges incident to some x ∈ GC, then N ≤ O(nk−1) and

N 2 k−1 k−1 ∑ βi ≤ (k − 1)!(2n) + o(n ). i=1 26 equiangular lines and spherical codes

0 Proof. Let x ∈ C and let C = px(Nαi (x)) be the normalized projection of the ⊥ 0 0 αi-neighbors of x onto span(x) . By Lemma 2.2.4, we see that C is a [−1, −β + 0 0 o(1)] ∪ {α1 + o(1),..., αk + o(1)}-spherical code for α − α2 β + α2 0 = j i ≤ ≤ 0 = i αj 2 for 1 j k, β 2 . 1 − αi 1 − αi

2 0 −αi 0 In particular α1 = 2 < 0. Now let ` be the largest integer such that α` < 0 and 1−αi 0 0 0 0 observe that C is, in particular, a [−1, α` + o(1)] ∪ {α`+1 + o(1),..., αk + o(1)}-code. If ` ≥ 2 then applying Theorem 2.5.4 by induction we obtain |C0| ≤ O(nk−`), which 0 trivially implies (i). Otherwise α2 ≥ 0 and applying Theorem 2.5.4 by induction we obtain 0 0 α2 k−1 k−1 |C | ≤ 1 + 0 (k − 2)!(2n) + o(n ). −α1 0 0 2 2 2 To verify (i), it sufﬁces to observe that 1 − α2/α1 = 1 + (α2 − αi )/αi = α2/αi ≤ 1/αi. N = d (x) M = M Now we derive the upper bound on β . Let Nβ (x)∪{x} be the | Gram matrix of Nβ(x) ∪ {x} and let v = (1, . . . , 1, βN) . Then using (i) we conclude | 2 2 0 ≤ v Mv ≤ ∑ Mij − β + o(1) N 1≤i,j≤N   k ≤ + ( + ( )) + ( + ( )) − 2 + ( ) 2 N 1 α1 o 1 N ∑ αj o 1 ∆αj  β o 1 N j=2 ≤ N 1 + (k − 1)!(2n)k−1 + o(nk−1) − β2 + o(1) N2,

≤ 1 ( − ) ( )k−1 + ( k−1) = ( k−1) which implies N β2 k 1 ! 2n o n O n . Finally, let −β1,..., −βN be the values of the β-edges incident to x and let N 2| w = β1,..., βN, ∑i=1 βi . Then

0 ≤ w| Mw

N !2 N 2 2 = − ∑ βi + ∑ βi + ∑ βi βj Mij i=1 i=1 1≤i,j≤N i6=j

N !2 N k ! 2 2 ≤ − ∑ βi + ∑ βi + ∑ αr ∑ βi βj + o(1) ∑ βi βj. i=1 i=1 r=2 i,j 1≤i,j≤N Mi,j=αr +o(1) Applying Cauchy–Schwarz, we obtain

N !2 N N 2 k−1 2 ∑ βi βj = ∑ βi ≤ N ∑ βi ≤ O(n ) ∑ βi . 1≤i,j≤N i=1 i=1 i=1 2.5 lines with many angles and related spherical codes 27

Furthermore, for 2 ≤ r ≤ k we have

1 N ≤ ( − )2 ≤ 2 − 0 ∑ βi βj ∆αr ∑ βi ∑ βi βj, 2 i,j i=1 i,j Ai,j =αr +o(1) Ai,j=αr +o(1) and thus we obtain N k−1 k−1 2 αr ∑ βi βj ≤ (k − 2)!(2n) + o(n ) ∑ βi . i,j i=1 Ai,j=αr +o(1)

N 2 Combining these inequalities and dividing by ∑i=1 βi yields the desired (ii): N 2 k−1 k−1 ∑ βi ≤ (k − 1)!(2n) + o(n ). i=1 Finally, we will need a new lemma to deal with what happens if the clique we ﬁnd via Ramsey’s theorem is an αi-clique for i ≥ 2.

Lemma 2.5.8. Let 2 ≤ i ≤ k and suppose X ∪ Y is a [−1, −β] ∪ {α1,..., αk}-spherical code with |Y| → ∞ as n → ∞, such that all edges incident to any y ∈ Y are αi-edges. Then |X| ≤ O(nk−1).

0 ⊥ Proof. Let X = pY(X) be the normalized projection of X onto span(Y) . By 0 0 0 0 Lemma 2.2.4, we have that X is a [−1, −β + o(1)] ∪ {α1 + o(1),..., αk + o(1)}-code for 0 αj − αi 0 β + αi αj = for 1 ≤ j ≤ k, β = . 1 − αi 1 − αi 0 0 0 Observe that αi−1 = (αi−1 − αi)/(1 − αi) < 0, so that X is, in particular, a [−1, αi−1 + 0 0 o(1)] ∪ {αi + o(1),..., αk + o(1)}-code and hence we may apply Theorem 2.5.4 by induction to conclude |X0| ≤ O(nk−i+1) ≤ O(nk−1).

We now have all of the necessary lemmas to ﬁnish the proof of Theorem 2.5.4.

Proof of Theorem 2.5.4. Suppose ﬁrst that α1 = 0. Let Q = MC − (α1 + o(1))J; by the subadditivity of the rank we have rk(Q) ≤ rk(MC) + rk(J) ≤ n + 1. Now ﬁx some x ∈ C, and let N = dβ(x) and β1,..., βN be the values of the β-edges incident to x. Using parts (i) and (ii) of Lemma 2.5.7, it follows that if i is the row corresponding to x in Q then

N k 2 ≤ (− − ( + ( )))2 + ( − + ( ))2 ∑ Qi,j ∑ βj α1 o 1 ∑ αr α1 o 1 ∆αr j6=i j=1 r=2 k ≤ (1 + o(1)) (k − 1)!(2n)k−1 + o(nk−1) + ∑ (k − 2)!(2n)k−1 + o(nk−1) r=2 ≤ 2(k − 1)!(2n)k−1 + o(nk−1). 28 equiangular lines and spherical codes

Noting that Q has 1 − (α1 + o(1)) = 1 − o(1) on the diagonal, we obtain

|C| |C| 2 2 2 k−1 k−1 tr(Q ) = ∑ Qi,i + ∑ ∑ Qi,j ≤ |C| 1 + 2(k − 1)!(2n) + o(n ) . i=1 i=1 j6=i

Thus applying Lemma 2.2.9 to Q yields

|C|2(1 − o(1))2 = tr(Q)2 ≤ tr(Q2)rk(Q) ≤ |C| 1 + 2(k − 1)!(2n)k−1 + o(nk−1) (n + 1).

Dividing by |C|(1 − o(1))2, we obtain the required |C| ≤ (k − 1)!(2n)k + o(nk). We will now prove the theorem for α1 > 0. Let t = log log n, let e → 0 sufficiently slowly as n → ∞ and suppose for sake of contradiction that |C| ≥ (1 + α1/β)(k − 1)!(2n)k + enk. Let m = d|C|/(k + 1)(k+1)t − 1e so that |C| > (k + 1)(k+1)tm. Regarding a γ-edge of C as an edge colored with the color γ, we deduce by Lemma 2.5.1 that there are some subsets X and Y of C so that |X| = m, |Y| = t 1 and (X, Y) is a monochromatic γ-pair for some γ ∈ {β, α1,..., αk}. Since t > β + 1 for n sufficiently large, Y cannot be a β-clique by Lemma 2.2.1 and hence (X, Y) cannot be a monochromatic β-pair. Hence it must be a monochromatic αi-pair. If 2 ≤ i ≤ k, then by Lemma 2.5.8 we conclude Ω(nk/(k + 1)(k+1)t) ≤ m ≤ O(nk−1), a contradiction for n large enough by our choice of t. Hence, (X, Y) must be a monochromatic α1-pair and hence C contains an α1-clique Y of size t. For each T ⊆ Y, let ST be the set of vertices x ∈ GC \ Y so that Nβ(x) ∩ Y = Y \ T. Now fix T ⊆ Y and let t1,..., t|T| be some ordering of the elements of T. For |T| each pattern of the form p ∈ [k] , let ST(p) consist of all x ∈ ST for which

hx, tii = αpi + o(1) for all i. ∗ 2 ∗ Define t = 4/ α1(β + α1) + o(1) and suppose first that t ≤ |T| < t. We 2 claim that ST(p) does not contain an α1-clique of size larger than 1/β + o(1) for |T| any p ∈ [k] . To that end, fix some z ∈ Y \ T and let X be an α1-clique in ST(p). Note that, for any x ∈ X, hx, zi < −β + o(1) and |T| ≥ t∗ > 4/(α + o(1))(β + α + o(1))2, so that we may apply Lemma 2.5.6 to X ∪ T ∪ {z} to conclude that |X| < 1/(β + α)2 + o(1) < 1/β2 + o(1). 0 (k+1)t (k+1)t 0 Now let m = d|ST(p)|/(k + 1) − 1e so that |ST(p)| > (k + 1) m . Then 0 0 0 0 by Lemma 2.5.1, ST(p) contains an (X , Y ) monochromatic pair with |X | = m and |Y0| = t. Since t > 1/β2 + o(1) for n large enough, Y0 cannot be a monochromatic 0 0 α1-clique or β-clique, and thus (X , Y ) is a monochromatic αi-pair for some 2 ≤ i ≤ k. Thus by Lemma 2.5.8, we conclude that m0 ≤ O(nk−1). Since this holds for all p, we obtain

|T| (k+1)t k−1 (k+2)t k−1 |ST| = ∑ |ST(p)| ≤ k (k + 1) O n ≤ (k + 1) O n . p∈[k]|T| 2.5 lines with many angles and related spherical codes 29

t Now suppose that T = Y and let p ∈ [k] \{(1, . . . , 1)}. We claim that SY(p) 2 does not contain an α1-clique of size larger than 1/(α2 − α1) + o(1). To that end, ﬁx an index j such that pj ≥ 2 and let X be an α1-clique in SY(p). Note that, for D E any x ∈ X, x, tj = αpj + o(1) ≥ α2 + o(1). Furthermore, for sufﬁciently large n, 2 we have t > 4/ α1(α2 − α1) + o(1). Therefore, we may apply Lemma 2.5.6, with 0 0 α = α1 + o(1) and γ = α2 + o(1), to X ∪ T ∪ {z}, where z = tj and T = Y\{tj}, to 2 conclude that |X| < 1/(α2 − α1) + o(1). 0 (k+1)t As above, let m = d|SY(p)|/(k + 1) − 1e and observe that by Lemma 2.5.1, 0 0 0 SY(p) contains an (X , Y ) monochromatic pair with |X | = m and |Y| = t. Since t is large enough, it cannot be a β or α1-pair, so it must be a monochromatic αi for some 2 ≤ i ≤ k, and hence by Lemma 2.5.8, we conclude that m0 ≤ O(nk−1). Thus we obtain

|T| (k+1)t k−1 (k+2)t k−1 ∑ |SY(p)| ≤ k (k + 1) O n ≤ (k + 1) O n . p∈[k]t \{(1,...,1)}

0 Finally, let p = (1, . . . , 1). and deﬁne X = pY(SY(1, . . . , 1)) to be the normalized ⊥ 0 projection of SY(1, . . . , 1) onto span(Y) . By Lemma 2.2.4 we have that X is a 0 0 0 [−1, −β + o(1)] ∪ {α1 + o(1),..., αk + o(1)}-spherical code for

0 αj − α1 0 β + α1 αj = for 1 ≤ j ≤ k, β = . 1 − α1 1 − α1 0 0 Since α1 = 0, we can apply the previous case of Theorem 2.5.4 to obtain |X | ≤ (k − 1)!(2n)k + o(nk). It follows that

|SY| = |SY(1, . . . , 1)| + ∑ |SY(p)| p∈[k]t \{(1,...,1)} ≤ (k − 1)!(2n)k + o(nk) + (k + 1)(k+2)tO nk−1

= (k − 1)!(2n)k + o(nk).

Noting that (k + 1)(k+2)t = o(n), we therefore obtain

[ [ S = |S | + S T Y T T⊆Y,|T|≥t∗ T⊂Y,|T|≥t∗ ≤ (k − 1)!(2n)k + o(nk) + 2t(k + 1)(k+2)tO nk−1

= (k − 1)!(2n)k + o(nk).

0 S 0 k Thus if we deﬁne G = GC\ T⊆Y,|T|≥t∗ ST then |G | ≥ (α1/β)(k − 1)!(2n) + k (e − o(1))n . Hence we can iterate the above procedure ` = α1/β + 1 times to obtain 0 2 disjoint α1-cliques Y1,..., Y` and a disjoint graph G of size at least (e − o(1))n . By having e → 0 slowly enough, we can apply Lemma 2.5.1 one more time to G0 to 30 equiangular lines and spherical codes

0 obtain a monochromatic α1-pair, which gives an additional α1-clique Y`+1 ⊆ G of size t. Note that by construction, the number of β-edges between Yi and Yj is at least t(t − t∗) = t2(1 − o(1)) for distinct i and j. But then we can apply Lemma 2.3.5 to obtain ` + 1 ≤ α1/β + 1 + o(1), a contradiction for n large enough.

2.6 concluding remarks

In this chapter, we showed that the maximum cardinality of an equiangular set of lines with common angle arccos α is at most 2n − 2 for ﬁxed α ∈ (0, 1) and large 1 n. Moreover, we proved that this bound is only attained for α = 3 and that we have an upper bound of 1.93n otherwise. In view of the result of Neumann [64, p. 498], it is not too surprising that lim supn→∞ Nα(n)/n should be biggest when 1/α is an odd integer. What is surprising, however, is that a maximum occurs at all and moreover that it happens when α is large. Indeed, the constructions of Ω(n2) equiangular lines have α → 0, and so one might a priori expect that lim supn→∞ Nα(n)/n should increase as α decreases. If α = 1/(2r − 1) for some positive integer r, an analogous construction as for α = 1 3 yields an equiangular set of rb(n − 1)/(r − 1)c lines with angle arccos(1/(2r − 1)). Indeed, consider a matrix with t = b(n − 1)/(r − 1)c blocks on the diagonal, each of size r, with 1 on the diagonal and −α off the diagonal; all other entries are α. One can show that this is the Gram matrix for a set of rt unit vectors in Rn. For n large enough and r = 2 [66] and r = 3 [74], it is known that this construction is optimal. This motivates the following conjecture, which was also raised by Bukh [16]. Conjecture 2.6.1. Let r ≥ 2 be a positive integer. Then, for sufﬁciently large n,

r(n − 1) N 1 (n) = + O(1). 2r−1 r − 1 After the results in this thesis were obtained, Jiang and Polyanskii [55] used our p √ framework to determine Nα(n) up to a constant, for all α > 1/(1 + 2 2 + 5). They conjectured that their result is true for all α and since it generalizes Conjec- ture 2.6.1, we formulate their conjecture below. 1−α Conjecture 2.6.2. Let α be ﬁxed and let kα be the smallest integer such that 2α is the largest eigenvalue of the adjacency matrix of some graph on kα vertices. If kα < ∞, then kα Nα(n) = n + O(1) and otherwise if kα = ∞, then Nα(n) = n + O(1). kα−1 We believe that the tools developed here should be useful to determine the asymptotics of Nα(n) for every ﬁxed α. If α is allowed to depend on n, then our methods work provided that α > Θ(log−1 n). The only place where this assumption is really necessary is our use of Ramsey’s theorem in order to obtain a large positive clique. However, it is conceivable that a large positive clique exists even when α < Θ(log−1 n), in which case our methods would continue to be effective. 2.6 concluding remarks 31

Remark 2.6.3. After the results in this thesis were obtained, Glazyrin and Yu [40] ( ) ( ) ≤ 2 + 4 + proved the ﬁrst universal bound on Nα n , showing that Nα n 3α2 7 n 2 for all α ≤ 1/3. We have also proved an upper bound of O(nk) for a set of lines attaining k prescribed angles. If the angles can tend to 0 together with n, however, this bound no longer applies and the general bound of O(n2k) by Delsarte, Goethals and Seidel [25] remains best possible. There are by now plentiful examples showing that for k = 1 their bound gives the correct order of magnitude, but no such constructions are known for other values of k. So it would be interesting to determine whether the bound of Delsarte, Goethals and Seidel is tight for k ≥ 2.

Acknowledgments. We thank Felix Dräxler, Peter Keevash, and Benny Sudakov for working with the author in order to obtain these results. We also thank Boris Bukh for useful comments.

3 EQUIANGULARSUBSPACES

3.1 introduction

Since equiangular lines have been studied extensively for many years, it is natural and interesting to consider analogous questions for k-dimensional subspaces. To this end, we must first understand the notion of angle between subspaces. We define the Grassmannian Gr(k, n) to be the set of all k-dimensional subspaces of Rn. Note that θ is the common angle between a pair of lines U, V ∈ Gr(1, n) if and only if cos θ = max hu, vi. u∈U,v∈V |u|=1,|v|=1 Generalizing this idea, given a pair of k-dimensional subspaces U, V ∈ Gr(k, n), we may recursively define the k principal angles 0 ≤ θ1 ≤ ... ≤ θk ≤ π/2 between U and V as follows: Choose unit vectors u ∈ U, v ∈ V that maximize hu, vi and define θ1 = arccos hu, vi. Now recursively define θ2,..., θk to be the principal angles between the (k − 1)-dimensional subspaces U0 = {u0 ∈ U : u0 ⊥ u} and V0 = {v0 ∈ V : v0 ⊥ v}. The geometric significance of principal angles are that they completely characterize the relative position of U to V, in the sense that if U0, V0 ∈ Gr(k, n) have the same principal angles as U, V, then there exists an orthogonal matrix Q such that U0 = {Qu : u ∈ U} and V0 = {Qu : u ∈ U}, see [84, Theorem 3]. It will be convenient for us to give another definition of principal angles that is more algebraic. Indeed, observe that a pair of lines U, V ∈ Gr(1, n) has common angle θ if and only if, when we choose any unit vectors u ∈ U, v ∈ V, we have (cos θ)2 = hu, vi2. More generally, we associate to a subspace U ∈ Gr(k, n), a representative n × k matrix U = (u1,..., uk) where u1,..., uk is any orthonormal basis of column vectors spanning U. Now given a pair of subspaces U, V ∈ Gr(k, n) with principal angles θ1,..., θk, one can show that cos θ1, . . . , cos θk are precisely | 2 2 the singular values of U V. In other words, (cos θ1) ,..., (cos θk) are precisely the eigenvalues of V|UU|V. Now that we understand angles between subspaces, we are ready to discuss the notion of equiangular subspaces. Note that one can consider equiangular sets of subspaces with respect to the principal angle θi for any fixed 1 ≤ i ≤ k. More generally, for any function d = d(θ1,..., θk) of the principal angles, we call a set of k-dimensional subspaces H ⊆ Gr(k, n) equiangular (with respect to d and having common angle α) if d(U, V) = α for all U 6= V ∈ H. Thus we may define and study d Nα (k, n), the maximum size of a set H ⊆ Gr(k, n) that is equiangular with respect d d to d and having common angle α, as well as N (k, n) = maxα Nα (k, n). We call a

33 34 equiangular subspaces

2 function d : Gr(k, n) → R an angle distance if d(U, V) ∈ {θ1(U, V),..., θk(U, V)} for all U, V ∈ Gr(k, n). If d satisﬁes d(U, V) = 0 iff U = V then we call d a proper distance. In section 3.2 we give examples of angle distances and prove a general upper d bound on Nα (k, n) for any angle distance d and α > 0, in particular improving and extending a result of Blokhuis [12], who studied the case d = θ1 and k = 2. Based on equiangular lines, we also give a lower bound construction of k-dimensional subspaces that are equiangular for any proper angle distance. We therefore conclude that for k ﬁxed and any proper angle distance d, Nd(k, n) = Θ(n2k) as n → ∞. In section 3.3, we discuss Nd(k, n) for some other well-studied distances d. In section 3.4, we conclude by stating some open problems, in particular discussing another generalization of equiangular lines known as equi-isoclinic subspaces.

3.2 angle distances

When trying to define the angle between two subspaces U, V ∈ Gr(k, n), one natural idea is to just take the minimum angle between any pair of vectors u ∈ U, v ∈ V. Since minimizing arccos hu, vi is equivalent to maximizing hu, vi, this idea gives exactly the first principal angle θ1 = θ1(U, V). This angle distance was first considered by Dixmier [30]. In [12], Blokhuis considered equiangular planes with respect to θ1 and proved that 2n + 3 Nθ1 (2, n) ≤ (3.1) α 4

provided that the common angle α > 0. This condition is necessary, since θ1(U, V) = 0 iff U and V share a nontrivial subspace, and so we could take infinitely many θ1 planes all sharing a fixed line, showing that N0 (2, n) = ∞. This is a troublesome property of θ1, because it shows that θ1 is not a proper distance and also that θ1 does not appeal to elementary geometric intuition. Indeed, consider a pair of 3 planes U, V in R . They will always share a line and hence will have θ1(U, V) = 0. However, one would intuitively ascribe the angle between them to be θ2(U, V). In view of this, it makes sense to define the minimum non-zero angle θF(U, V) = min{θi(U, V) : θi(U, V) > 0}. θF was first considered by Friedrichs [37] and it is a proper angle distance. Deutsch [27] gives applications of θ1 and θF to the rate of convergence of the method of cyclic projections, existence and uniqueness of abstract splines, and the product of operators with closed range. Another proper angle distance is the maximum angle θk, first considered by Krein, Krasnoselski, and Milman [62]. It was used by Asimov [8] for his “Grand Tour,” a method for visualizing high dimensional data by projecting to various two- dimensional subspaces and showing these projections sequentially to a human. θk was also considered by Conway, Hardin, and Sloane [22] in their paper on packing subspaces in Grassmannians. d For any angle distance d and α > 0, we give an upper bound on Nα (k, n) on the 2k order of n , extending Gerzon’s bound in eq. (2.1). In the case d = θ1 and k = 2, 3.2 angle distances 35 this improves Blokhuis’ bound in eq. (3.1). The proof is based on the polynomial method, which was also the main tool in [12]. Theorem 3.2.1. Let k, n ∈ N with k ≤ n, let d be an angle distance on Gr(k, n) and let α > 0. Then (n+1) + k − 1 Nd(k, n) ≤ 2 . α k

Proof. Let {U1,..., Um} ⊆ Gr(k, n) be a set of subspaces such that d(Ui, Uj) = α for all i 6= j and for each Ui, let Ui = (u1,..., uk) be a representative n × k matrix where u1,..., uk is any orthonormal basis of column vectors spanning Ui. Observe that for any i 6= j, since α = d(Ui, Uj) is a principal angle between Ui and 2 Uj, we have, as per the discussion in section 3.1, that (cos α) is an eigenvalue of | | 2 | | Ui UjUj Ui. Thus if we deﬁne λ = (cos α) then we have det Ui UjUj Ui − λIk = 0, where Ik is the k × k identity matrix. Now let S = {X ∈ Rn×n : X| = X} be the set of all symmetric n × n matrices and deﬁne functions f1,..., fm : S → R by λtr(X) f (X) = det U|XU − I . i i i k k

| | Since tr(UjUj ) = tr(Uj Uj) = tr(I) = k, we conclude that   k | (1 − λ) if i = j fi(UjUj ) = 0 if i 6= j.

Moreover, note that λ 6= 1 since α 6= 0. It therefore follows that f1,..., fm are m linearly independent. Indeed, if ∑i=1 ci fi = 0 for some c1,..., cm ∈ R, then for all m | k j we have 0 = ∑i=1 ci fi(UjUj ) = cj(1 − λ) , which implies cj = 0. (n+1)+k−1 ( 2 ) Thus it sufﬁces to show that f1,..., fm live in a space of dimension k . To that end, recall that a multivariable polynomial f : Rt → R is called homogeneous of degree k if it is a linear combination of monomials of degree k, and that the linear t+k−1 space of such polynomials has dimension ( k ). For any X ∈ S , we let Xa,b denote the entry in position a, b of the matrix X, so that S may be parametrized n+1 by the ( 2 ) variables {Xa,b : 1 ≤ a ≤ b ≤ n} living on or above the diagonal and hence we may think of the functions fi as polynomials in these variables. Now | λtr(X) observe that for any i and X ∈ S , every entry of the k × k matrix Ui XUi − k Ik is a homogeneous polynomial of degree 1 in the variables {Xa,b : 1 ≤ a ≤ b ≤ n}. It follows from the deﬁnition of the determinant that fi(X) is a homogeneous n+1 polynomial of degree k in these variables. Since there are ( 2 ) such variables, the space of all homogeneous polynomials of degree k in these variables has dimension (n+1)+k−1 ( 2 ) k , completing the proof. 36 equiangular subspaces

To obtain lower bounds for this problem, it is natural to start with a construction of many equiangular lines and then try to combine them to make k-dimensional subspaces. Recall that N(n) is the maximum size of a set of equiangular lines in Rn. In the following, we make use of the Frobenius inner product hA, Bi = tr(A|B) for n × n real-valued matrices A, B. Theorem 3.2.2. For any k, n ∈ N with k ≤ n, there exists a set H ⊆ Gr(k, kn) with |H| = N(n)k and α ∈ (0, π/2) such that for all U, V ∈ H, the principal angles between U and V all lie in the set {0, α}.

Proof. Let L ⊆ Gr(1, n) be an equiangular set of lines with |L| = N(n), and let α ∈ (0, π/2) be the common angle of any pair of lines in L. Now let C be the set of vectors obtained by choosing a unit vector along each line in L, and observe that hu, vi2 = (cos α)2 for all u 6= v ∈ C. k Now let e1,... ek be the standard basis in R , and observe that for all u, v ∈ C, we have  D E hu vi i = j | | | | | | , if eiu , ejv = tr(uei ejv ) = (ei ej)(u v) = 0 if i 6= j.

| kn Now observe that for any i and u ∈ C, eiu can be viewed as a vector in R and | | thus if we let u1,..., uk ∈ C, then e1u1 ,..., ekuk can be viewed as orthonormal kn vectors in R and hence deﬁne a subspace Wu1,...,uk in Gr(k, kn). Furthermore, for all u1,..., uk, v1,..., vk ∈ C, if we let U be the kn × k matrix with column vectors | | | | e1u1 ,..., ekuk and let V be the kn × k matrix with column vectors e1v1 ,..., ekvk ,

then U is a representative matrix for Wu1,...,uk and V is a representative matrix for

Wv1,...,vk . Now we compute that D E D E  u , v if i = j | | | i j (U V)i,j = eiui , ejvj = 0 if i 6= j,

and hence D E2  u , v if i = j | | i j (V UU V)i,j = 0 if i 6= j. Thus the eigenvalues of V|UU|V lie in the set {1, cos(α)2} and so the principal

angles between Wu1,...,uk and Wv1,...,vk lie in the set {0, α}. Letting H = {Wu1,...,uk : k u1,..., uk ∈ C} and observing that |H| = N(n) completes the proof. Next we show that the construction above is equiangular for any proper angle distance d, and hence obtain the following corollary. Corollary 3.2.3. Let d be a proper angle distance and let k ∈ N be ﬁxed. Then

Nd(k, n) = Θ(n2k) as n → ∞. 3.3 other distances 37

Proof. Theorem 3.2.1 immediately gives the upper bound Nd(k, n) ≤ O(n2k). For the lower bound, let α ∈ (0, π/2) and H ⊆ Gr(k, kn) be given by Theorem 3.2.2. Observe that for all U 6= V ∈ H, the principal angles between U and V cannot all be 0, and thus θF(U, V) = θk(U, V) = α. Moreover, observe that since d is a proper angle distance, we have θF ≤ d ≤ θk. Thus d(U, V) = α for all U 6= V ∈ H. De Caen’s bound eq. (2.2) implies that N(n) ≥ Ω(n2) and so we obtain

Nd(k, kn) ≥ |H| = N(n)k ≥ Ω(n2k).

Thus we conclude Nd(k, n) ≥ Ω(n2k).

3.3 other distances

Besides angle distances, there are several other natural distance functions that are considered in geometry, statistics, and applied problems, see e.g. [32]. Let n U, V ∈ Gr(k, n) be k-dimensional subspaces of R with principal angles θ1,..., θk. If one considers the Grassmanian Gr(k, n) as a manifold, one may compute (see [84, Theorem 8]) that the geodesic distance is q 2 2 dG(U, V) = θ1 + ... + θk . In the context of packing subspaces, Conway, Hardin, and Sloane [22] consider the geodesic distance, the maximum principal angle, as well as the chordal distance defined by q q 2 2 | | dC(U, V) = (sin θ1) + ... + (sin θk) = k − tr(V UU V). Also in the context of packing subspaces, Dhillon, Heath, Strohmer, and Tropp [28] consider the first principal angle (spectral distance), as well as the Fubini-Study distance defined by

k ! | dFS(U, V) = arccos ∏ cos θi = arccos |det U V|. i=1

For a subspace U ∈ Gr(k, n), we deﬁne the orthogonal complement U⊥ = {v ∈ Rn : v ⊥ u for all u ∈ U}. The following lemma shows us that the nonzero principal angles between subspaces are the same as the nonzero principal angles between their orthogonal complements. Lemma 3.3.1. For any U, V ∈ Gr(k, n), the nonzero principal angles between U⊥ and V⊥ are the same as the nonzero principal angles between U and V.

Proof. Observe that UU| is an orthogonal projection onto U and U⊥(U⊥)| is an ⊥ ⊥ ⊥ orthogonal projection onto U , so that UU| + U (U )| = In where In is the n × n identity matrix. Thus

| | | ⊥ ⊥ | | ⊥ ⊥ | U VV U = U (In − V (V ) )U = Ik − U V (V ) U 38 equiangular subspaces

and

⊥ | ⊥ ⊥ | ⊥ ⊥ | | ⊥ ⊥ | | ⊥ (V ) U (U ) V = (V ) (In − UU )V = In−k − (V ) UU V .

Since it is well known that for any A, B the matrices AB and BA have the same nonzero eigenvalues with the same multiplicity, we have that U|V⊥(V⊥)|U has the same nonzero eigenvalues as (V⊥)|UU|V⊥, and therefore U|VV|U has the same eigenvalues as (V⊥)|U⊥(U⊥)|V⊥ except for eigenvalues of 1. Hence the principal angles between U and V are the same as the principal angles between U⊥ and V⊥, except for angles of 0.

Now let d be one of the proper distances discussed in this paper, and observe that principal angles of 0 don’t affect d. Thus using Lemma 3.3.1, we have that d(U⊥, V⊥) = d(U, V) for all U, V ∈ Gr(k, n). We therefore conclude that ⊥ ⊥ U1,..., Um ∈ Gr(k, n) are equiangular with respect to d iff U1 ,..., Um ∈ Gr(n − k, n) are equiangular with respect to d, and hence that

Nd(k, n) = Nd(n − k, n).

Thus, for the purposes of studying Nd(k, n), it will sufﬁce for us to consider the case k ≤ n/2. Conway, Hardin, and Sloane [22] give some reasons why they consider the chordal distance dC to be the best deﬁnition for packings, in particular observing that the Grassmanian Gr(k, n) with the chordal distance can be isometrically D n+1 embedded onto a sphere in R for D = ( 2 ) − 1, by mapping a subspace U to the projection matrix UU| and using the Frobenius inner product tr(A|B). Since an equidistant set (simplex) in RD has size at most D + 1, they conclude that

n + 1 NdC (k, n) ≤ , 2 generalizing Gerzon’s bound eq. (2.1). For a lower bound, given a set of m k- dimensional subspaces U1,..., Um ∈ Gr(k, n) equiangular with respect to dC, ob- 0 serve that by adding a new dimension and deﬁning Ui = span(Ui, en+1), we obtain a set of m (k + 1)-dimensional subspaces in Gr(k + 1, n + 1) which is equiangular dC dC with respect to dC. Thus N (k + 1, n + 1) ≥ N (k, n) for all k ≤ n and so, using the assumption k ≤ n/2 together with eq. (2.2), we obtain

NdC (k, n) ≥ NdC (1, n − k + 1) = N(n − k + 1) = Ω(n2).

Additionally, for a prime p such that a Hadamard matrix of order (p + 1)/2 exists, (p+1) Calderbank, Hardin, Rains, Shor, and Sloane [21] give a construction of 2 sub- p spaces of dimension (p − 1)/2 in R which are equiangular with respect to dC, so that p + 1 NdC ((p − 1)/2, p) = . 2 3.4 concluding remarks 39

For the Fubini-Study distance dFS, we will need some definitions from multilin- ear algebra, see e.g. [83] for reference. Let u ∧ v denote the wedge product between n Vk n n u, v ∈ R . Let (R ) = {u1 ∧ ... ∧ uk : u1,..., uk ∈ R } denote the kth exte- n Vk n n rior power of R and note that n (R ) = (k). We shall use the Plücker embedding of Gr(k, n) into the projective space of lines over Vk(Rn), defined as follows. Given a subspace U ∈ Gr(k, n) with u1,..., uk being an orthonormal basis of col- Vk n umn vectors of U, we define φ(U) = u1 ∧ ... ∧ uk ∈ (R ). One can compute that hφ(U), φ(V)i = det(U|V) defines an inner product between φ(U) and φ(V). Therefore, given a set of subspaces U1,..., Um ∈ Gr(k, n) equiangular with respect to dFS, we have that φ(U1),..., φ(Um) are a set of vectors such that if we take a line along each vector, we obtain a set of equiangular lines in Vk(Rn). Thus using eq. (2.1), we conclude (n) + 1 NdFS (k, n) ≤ k . 2 Actually, the Plücker embedding gives an embedding into an algebraic variety over Vk(Rn) defined by the so-called Plücker relations, and so it conceivable that this can be used to obtain a better upper bound. If a matching lower bound construction exists, finding it seems difficult since it would, in particular, yield a new construc- 2 N n tion of Ω(N ) equiangular lines in R , for N = (k). We do not know anything about equiangular subspaces for the geodesic distance dG, as well as other distances which cannot be written in terms of polynomial expressions of cos θ1, . . . , cos θk. This is not surprising, since all of the above upper bounds are essentially proven via the polynomial method. It would, therefore, be interesting to find other methods for proving such upper bounds.

3.4 concluding remarks

d 2k In section 3.2 we give an upper bound on Nα (k, n) of the order n for any angle distance d and α > 0, but are only able to give a corresponding lower bound when d is a proper angle distance. It would therefore be interesting to give lower bound constructions (with common angle α > 0) on the order of n2k for angle distances that are not proper, in particular for the minimum angle θ1. Moreover, if n k → ∞ then even for proper angle distances d, Corollary 3.2.3 still leaves open the correct asymptotic dependence of Nd(k, n) on k. In section 3, we remark that the polynomial method does not seem to work for distances such as the geodesic distance dG, and so it would be interesting to ﬁnd new methods which give upper bounds for such cases. It would also be interesting to obtain lower bound constructions for the Fubini-study distance dFS, and establish the correct order of magnitude for NdFS (k, n). Another approach to generalizing equiangular lines is, given a set H ⊆ Gr(k, n), to require that H is equiangular with respect to θi for all 1 ≤ i ≤ k. If we further require that θ1 = ... = θk, we arrive at the notion of equi-isoclinic subspaces. Equivalently, a family of subspaces H ⊆ Gr(k, n) is equi-isoclinic if there exists 40 equiangular subspaces

λ ∈ [0, 1) such that V|UU|V = λI for all U 6= V ∈ H. Lemmens and Seidel [65] deﬁned and studied v(k, n), the maximum number of k-dimensional equi-isoclinic subspaces in Rn. They gave a construction based on equiangular lines showing that v(k, kn) ≥ v(1, n) and generalized Gerzon’s bound in eq. (2.1), obtaining n+1 k+1 v(k, n) ≤ ( 2 ) − ( 2 ) + 1. Note that for n k → ∞, these bounds together with the fact that v(1, n) = N(n) ≥ Ω(n2) show that

n2 Ω ≤ v(k, n) ≤ O(n2). k2 It would be interesting to close this gap and determine the correct asymptotic dependence of v(k, n) on k.

Acknowledgments. We thank Benny Sudakov for working with the author in order to obtain these results. 4 ORTHONORMALREPRESENTATIONSOF H -FREEGRAPHS

4.1 introduction

Given a graph G, a map f : V ( G ) → R d is called an orthonormal representation of G (in R d ) if | | f ( u ) | | = 1 for all u ∈ V ( G ) and h f ( u ) , f ( v ) i = 0 for all distinct u , v ∈ V ( G ) such that u v ∈/ E ( G ). Note that every graph G on n vertices has at least one orthonormal representation, since we may assign each vector to a corresponding orthonormal basis vector in R | G | . Given an orthonormal representation f of a graph G with vertex set [ n ], we define M f to be the Gram matrix of the vectors f ( 1 ) ,..., f ( n ), so that ( M f ) i , j = h f ( i ) , f ( j ) i. The concept of orthonormal representations goes back to a seminal paper of Lovász [67], who used them to define a graph parameter known as the Lovász ϑ- function. The ϑ-function of a graph G has several equivalent definitions. Here we list the ones that we shall use later. Definition 4.1.1. Let G be a graph with vertex set [ n ]. The ϑ-function of G, denoted ϑ ( G ), can be defined in the following ways, which are shown to be equivalent in [67]. 1. ϑ ( G ) is the maximum, over all orthonormal representations f of the complement graph G, of the largest eigenvalue of the Gram matrix M f .

2. ϑ ( G ) is the maximum of 1 − λ 1 ( A ) / λ n ( A ), over all n × n real sym- 1 metric matrices A such that A i , j = 0 if i j ∈ E ( G ) or i = j .

3. ϑ(G) is the minimum, over all orthonormal representations f of G and all −2 unit vectors x, of maxv∈V(G) hx, f (v)i . 4. ϑ(G) is the maximum, over all orthonormal representations f of the comple- 2 ment graph G and all unit vectors x, of ∑v∈V(G) hx, f (v)i . Lovász originally introduced the notion of the ϑ-function in order to bound and compute the Shannon capacity of a graph, and since then, the combinatorial and algorithmic applications of the Lovász ϑ-function have been studied extensively, see e.g. Knuth [58]. Given a graph G, let us deﬁne the minimum semideﬁnite rank of G, denoted msr(G), to be the minimum d such that there exists an orthonormal representation of G in Rd. Note that msr(G) can be seen as a vector generalization of the

1 In [67], Lovász forgets to include the assumption that A is symmetric and Ai,i = 0 for all i to his statement of Theorem 6, but it is clear that this is what he intended.

41 42 orthonormal representations of h-free graphs

chromatic number of G, see [51]. Indeed, by assigning a standard basis vector of Rχ(G) to each vertex of a given color, one can see that msr(G) ≤ χ G. In the same paper where he introduced the ϑ-function, Lovász [67] implicitly showed that

ϑ(G) ≤ msr(G).

Various notions of the minimum rank of a graph have been studied in the lit- erature, see Fallat and Hogben [35] for a survey. Note that an equivalent way to define the minimum semidefinite rank of a graph G is as the minimum rank of a positive semidefinite matrix M such that Mi,i = 1 for all i and Mi,j = 0 if ij ∈/ E(G). Note that if we drop the positive semidefinite assumption, we arrive at the notion of min-rank which has applications in theoretical computer science [19, 43]; see Chapter 5 for more information.

4.1.1 A geometric problem of Lovász

One very interesting application of the Lovász ϑ-function is to the following geometric problem posed by Lovász and ﬁrst studied by Konyagin [59]. n What is the maximum ∆n, of the length ∑i=1 xi , over all d and all d unit vectors x1,..., xn ∈ R such that among any three, there is at least one pair of orthogonal vectors?

Konyagin [59] gave upper and lower bounds on ∆n, in particular showing that 3 2/3 ∆n ≤ 2 n . Then Kashin and Konyagin [57] improved the lower bound to within a logarithmic factor of the upper bound, and ﬁnally, Alon [1] was able to give 2/3 an asymptotically tight construction to conclude that ∆n = Θ(n ). Note that

if we deﬁne L(G) to be the maximum of ∑v∈V(G) f (v) over all orthonormal representations f for G, then the above problem is equivalent to asking for the maximum of L(G) over all triangle-free graphs G on n vertices. The following theorem connects L(G) to ϑ(G) and ϑ G. Theorem 4.1.2. For any graph G on n vertices, we have

n q p ≤ L(G) ≤ nϑ G . ϑ(G) q Moreover, if G is vertex-transitive, then L(G) = nϑ G.

For graphs G, H we say that G is H-free if G does not contain a copy of H as a subgraph. Generalizing from a triangle to an arbitrary H, let us now deﬁne λ(n, H) to be the maximum value of ϑ G over all H-free graphs G on n vertices. Although in this thesis we only study λ(n, H), we remark that roughly speaking, Theorem 4.1.2 would allow one to translate these results to the corresponding geometric problem of ﬁnding the maximum of L(G) over all H-free graphs G on n vertices, especially because the constructions we discuss are roughly vertex-transitive. Indeed, 4.1 introduction 43

for H = K3, Konyagin’s argument for the upper bound on ∆n can be adapted to 1/3 obtain λ(n, K3) ≤ O n , and since Alon’s construction for the lower bound on 1/3 ∆n is vertex-transitive, Theorem 4.1.2 implies that λ(n, K3) ≥ Ω n , so that we 1/3 have λ(n, K3) = Θ(n ). Generalizing to larger cliques, it is known that

1−O(1/ log t) 1−2/t Ω n ≤ λ(n, Kt) ≤ O n , where Alon and Kahale [5] proved the upper bound and Feige [36] proved the lower bound. Another way to generalize forbidding a triangle is to forbid longer cycles. Indeed, Alon and Kahale [5] also showed that for any t, if G is a graph on n vertices having no odd cycle of length at most 2t + 1, then θ G ≤ 1 + (n − 1)1/(2t+1). Our ﬁrst contribution is a generalization of this upper bound to graphs that have no cycle of length exactly 2t + 1.

1/(2t+1) Theorem 4.1.3. For all n, t ≥ 1 we have λ(n, C2t+1) ≤ O t n .

Alon and Kahale also noted that their result is tight via a modiﬁcation (see e.g. section 3, example 9 of [63]) of the construction of Alon [1]. The key properties that make such a construction useful are that it has many edges and that it has p an optimal spectral gap, which is to say that |λi(A)| ≤ O λ1(A) for 2 ≤ i ≤ n, where A is the adjacency matrix of the graph and λ1(A) ≥ ... ≥ λn(A) are its eigenvalues. Indeed, graphs with an optimal spectral gap and many edges have an adjacency matrix with a large ratio of |λ1(A)/λn(A)|, which by Deﬁnition 2 of the ϑ-function leads to a good lower bound for θG. For a graph H, the Turán number ex(n, H) is the maximum number of edges in an H-free graph G on n vertices. For bipartite H such as C4, C6, C10, K2,t and Kt,(t−1)!+1, there are known constructions of H-free graphs G that attain good lower bounds for the Turán number, i.e. |E(G)| ≥ Ω(ex(n, H)). Interestingly, these constructions also happen to have optimal spectral gaps, so it follows from the previous discussion that in such cases, r ! ex(n, H) λ(n, H) ≥ ϑG ≥ Ω . n We therefore obtain the following lower bounds. Theorem 4.1.4. Let n ≥ 1.

1/t 1. For all t ∈ {4, 6, 10} ∪ (2N + 1), we have λ(n, Ct) ≥ Ω n .

1/4 1/4 2. For all t ≥ 2, we have λ (n, K2,t) ≥ Ω t n .

1 (1−1/t) 3. For all t ≥ 3, we have λ n, Kt,(t−1)!+1 ≥ Ω n 2 . 44 orthonormal representations of h-free graphs

Since the Turán number can sometimes provide a lower bound for λ(n, H), one might wonder if it can also provide an upper bound. If H is a graph such that we can remove a vertex to obtain a tree and we have ex(n, H) ≤ O(n1+α) for some α > 0, then we are able to obtain such an upper bound. Theorem 4.1.5. Let h ≥ 1 and let H be a connected graph on h vertices with vertex set V(H) = T ∪ {v} where H[T] is a tree. Furthermore, suppose that there exist c, α with 0 < α ≤ 1 and c ≥ 1 such that ex(n, H) ≤ cn1+α for all n ≥ 1. Then for all n ≥ 1, it holds that √ ch λ(n, H) ≤ 20 nα/2. α

Now deﬁne θt,s to be the graph consisting of s internally disjoint paths of length t between a pair of vertices, and note that in particular θt,2 = C2t and θ2,s = K2,s. Since θt,s consists of a tree together with an additional vertex, we may use Theorem 4.1.5 together with known upper bounds on Turán numbers to obtain the following corollary.

2 1−1/(2t) 1/(2t) Corollary 4.1.6. Let n ≥ 1. For all t, s ≥ 2, we have λ(n, θt,s) ≤ O t s n . In particular, for all t ≥ 2, we have

2 1/(2t) 3/4 1/4 λ(n, C2t) ≤ O t n λ(n, K2,t) ≤ O t n .

1/(2t) Remark 4.1.7. The upper bound for λ(n, C2t) can be improved to O t n by making use of the proof technique used in Theorem 4.1.3, see the appendix for details.

Note that the lower bounds for Ct and K2,t given in Theorem 4.1.4 have corresponding upper bounds via Theorem 4.1.3 and Corollary 4.1.6, which are tight up to the constants depending on t. Unfortunately, since Kt,(t−1)!+1 for t ≥ 3 is not a tree together with a vertex, we are only able to obtain a weak upper bound in this case.

Theorem 4.1.8. For all s ≥ t ≥ 2, there exists a constant ct,s such that

1−2/t+1/(t 2t−1) λ(n, Kt,s) ≤ ct,s n .

4.1.2 Almost orthogonal vectors

Upon hearing about the results of Kashin and Konyagin [57] towards Lovász’s problem, Erd˝osasked the following related question (see Nešetˇriland Rosenfeld [73] for a historical summary):

What is the maximum, α(d), of the number of vectors in Rd such that among any three distinct vectors there is at least one pair of orthogonal vectors? 4.1 introduction 45

Rosenfeld [79] called such vectors almost orthogonal. By taking two copies of each of the vectors from a basis in Rd, we obtain 2d almost orthogonal vectors. Erd˝os believed that a construction with more than 2d vectors does not exist, and indeed Rosenfeld showed that α(d) = 2d (see Deaett [24] for a short and nice proof that is slightly more general). After his initial question was resolved, Erd˝osasked what happens if we replace 3 by a larger integer t. Füredi and Stanley [39] defined α(d, t) to be the maximum number of vectors in Rd such that, among any t + 1 distinct vectors, some pair is orthogonal. By considering t orthogonal bases in Rd, we obtain α(d, t) ≥ tk, and Erd˝osasked whether equality holds. Füredi and Stanley proved that it does not always hold by showing that α(4, 5) ≥ 24, and conjectured that there exists a constant c such that α(d, t) < (dt)c. This conjecture was later also proven false by Alon and Szegedy [7], who showed that for some constant δ > 0 and t large δ log t enough, α(d, t) ≥ d log log t . One can see that Erd˝os’squestion is almost equivalent to asking for the minimum of msr(G) over all Kt+1-free graphs G on n vertices. The difference is that Erd˝oswas asking for the vectors to be distinct, while an orthonormal representation of a graph may label multiple vertices with the same vector. Nonetheless, we shall define and study ρ(n, H), the minimum of msr(G) over all H-free graphs G on n vertices. Some further motivation for studying ρ(n, H) comes from Pudlák [76], who, inspired by questions in circuit complexity, studied the min-rank and minimum semidefinite rank of graphs without a cycle of given length. More recently, Haviv [48, 49] used the probabilistic method in order to construct graphs with large min-rank and whose complements are H-free. We note that the aforementioned results now take the form ρ(n, K3) = dn/2e, and log log t δ log t ρ(n, Kt+1) ≤ n for some constant δ > 0, t sufficiently large, and an infinite number of values of n. Surprisingly, for t fixed and n large, it seems that the best known lower bound on d+t ρ(n, Kt+1) is just what one gets from Ramsey theory: If n > ( t ) ≥ R(d + 1, t + 1) then any Kt+1-free graph on n vertices has an independent set of size d + 1, and d d+t t therefore cannot have an orthonormal representation in R . Since ( t ) = O d , 1/t we may conclude that ρ(n, Kt+1) ≥ Ω(n ). Making use of Alon and Kahale’s 1−2/k result [5] that λ(n, Kk) ≤ O(n ), we give a small improvement to this lower bound. Theorem 4.1.9. There exists a constant δ > 0 such that for all t ≥ 3 and n ≥ 1, 3/t ρ(n, Kt) ≥ δn . In the previous section, we saw that another way to generalize a question for triangle-free graphs is to forbid a longer cycle. Pudlák [76] (Theorem 10) gave a case-based proof showing that there exists c > 0 such that ρ(n, C5) ≥ cn. Taking 46 orthonormal representations of h-free graphs

t − 1 copies of each vector of an orthonormal basis in Rd gives an orthonormal representation of the graph consisting of d cliques of size t − 1, which implies

ρ(n, Ct) ≤ dn/(t − 1)e. Inspired by Erd˝os,we may ask if equality holds. Unlike before, we show that the answer turns out to be yes, in particular improving and generalizing Pudlák’s result.

Theorem 4.1.10. For all t ≥ 3, n ∈ N we have ρ(n, Ct) = dn/(t − 1)e. Indeed, this follows from the following more general result, which holds for all connected graphs H containing a vertex such that removing it, we obtain a tree. Theorem 4.1.11. Let t ≥ 1 and let H be a connected graph with vertex set V(H) = T ∪ {v} where H[T] is a tree on t vertices. Then for all n ≥ 1, ρ(n, H) = dn/te. Remark 4.1.12. Our definition of msr(G) differs from the minimum semidefinite rank defined by Deaette [24]. Indeed, the representations f : V(G) → Cd that he considers map into complex d-dimensional space, are allowed to map vertices to the 0 vector, and most importantly, must satisfy that h f (u), f (v)i 6= 0 if and only if uv ∈ E(G). The last condition defines a faithful representation, as studied by Lovász, Saks, and Schrijver [69]. Nevertheless, Theorems 4.1.9 to 4.1.11 may be adapted to work with these alternate assumptions. We prove our results in the next two sections. We first prove Theorems 4.1.9 to 4.1.11 in Section 4.2, and then proceed to prove the remaining results in Sec- tion 4.3. The final section of this chapter contains some concluding remarks.

4.2 minimum semidefinite rank for H-free graphs

In order to study the minimum semideﬁnite rank of a graph, we recall the useful Lemma 2.2.9 from Chapter 2 which states that for any symmetric real matrix M, rk( M) ≥ tr( M)2 /tr( M2 ). Now we are ready to prove Theorem 4.1.9 and Theo- rem 4.1.11, from which Theorem 4.1.10 will immediately follow.

Proof of Theorem 4.1.9. Let δ be a sufﬁciently small constant. We proceed by induction on t. For t = 3 we know that ρ(n, K3 ) = dn/2e ≥ δn. d Now let t ≥ 3 and let G be a Kt+1-free graph on n vertices. Let f : V (G) → R d be an orthonormal representation of G in R with M = M f being the corresponding Gram matrix. We will make use of Lemma 2.2.9. To this end, we shall upper bound tr( M2 ). Observe that   2 2 tr M2 = ∑  ∑ h f (u), f (w)i + ∑ h f (u), f (w)i  u∈V (G) w∈/ N (u) w∈ N (u)   2 = ∑ 1 + ∑ h f (u), f (w)i . u∈V (G) w∈ N (u) 4.2 minimum semidefinite rank for H-free graphs 47

Now ﬁx u ∈ V (G) and note that G[ N (u)] is Kt-free. Thus by the induction 3/t hypothesis, we have d ≥ ρ(| N (u)|, Kt ) ≥ δ| N (u)| . Since Alon and Kahale 1−2/t [5] showed that there exists a constant c such that λ(n, Kt ) ≤ c n , we have via Deﬁnition 4 of the ϑ-function that

2 t ∑ h f (u), f (w)i ≤ ϑ G[N(u)] ≤ λ(|N(u)|, Kt) ≤ λ (d/δ) 3 , Kt w∈N(u) 2 t 1− t ≤ c (d/δ) 3

t−2 = c(d/δ) 3 . Therefore, we conclude that tr M2 ≤ n 1 + c(d/δ)(t−2)/3 . Clearly tr(M) = n and rk(M) ≤ d, so applying Lemma 2.2.9 and dividing by n yields n ≤ d 1 + c(d/δ)(t−2)/3 = d + cδ−(t−2)/3d(t+1)/3 ≤ (d/δ)(t+1)/3 for δ a bit smaller than 1/c. Thus d ≥ δn3/(t+1) and since G and f were arbitrary, we conclude 3/(t+1) ρ(n, Kt+1) ≥ δn . Proof of Theorem 4.1.11. Let d = dn/te and G be a graph consisting of d cliques of size t. Since H is connected and has t + 1 vertices, G is clearly H-free. By assigning d the standard basis vector ei ∈ R to each vertex in the i-th clique, we obtain an orthonormal representation of G in Rd, so that we conclude ρ(n, H) ≤ d = dn/te. For the lower bound, let d = ρ(n, H) and let G be an H-free graph on n vertices that has an orthonormal representation f in Rd with corresponding Gram matrix M = M f . Following the approach of Deaette [24], we will use Lemma 2.2.9. To this end, we note that as in the proof of Theorem 4.1.9   2 tr M2 = ∑ 1 + ∑ h f (u), f (w)i . u∈V(G) w∈N(u)

Now ﬁx u ∈ V(G) and observe that since G has no copy of H, G[N(u)] has no copy of some tree on t vertices. It is well-known that in this case, χ(G[N(u)]) ≤ t − 1, see e.g. corollaries 1.5.4 and 5.2.3 of Diestel [29]. Thus we can partition N(u) into t − 1 independent sets B1,..., Bt−1. Since { f (w) : w ∈ Bi} is an orthonormal h ( ) i2 ≤ || ||2 set of vectors, we have by Parseval’s inequality that ∑w∈Bi f w , v v for any v ∈ Rd. In particular for v = f (u), we therefore have

t−1 t−1 ∑ h f (u), f (w)i2 = ∑ ∑ h f (u), f (w)i2 ≤ ∑ || f (v)||2 = t − 1, w∈N(u) i=1 w∈Bi i=1 and thus tr M2 ≤ ∑ (1 + t − 1) = nt. v∈V(G) 48 orthonormal representations of h-free graphs

Clearly we have tr(M) = n and rk(M) ≤ d, so that by Lemma 2.2.9 we obtain n2 ≤ dnt. Thus we conclude that d ≥ n/t and so ρ(n, H) = d ≥ dn/te, as desired.

4.3 lovász ϑ-function for H-free graphs

Proof of Theorem 4.1.2. Let f be an orthonormal representation of G that attains the maximum in the deﬁnition of L(G), and denote its Gram matrix by M f . We have that 2

L(G)2 = f (v) = h f (u), f (v)i ∑ ∑ v∈V (G) u,v∈V (G) | = (1, . . . , 1) M f (1, . . . , 1) ≤ nϑ G ,

where the last inequality follows from Deﬁnition 1 of the ϑ-function. For the other direction, let f ∗ be an orthonormal representation of G and x be a unit vector that together attain the minimum in Deﬁnition 3 of θ (G). We therefore − have that ϑ(G) ≥ hx, f ∗ (v)i 2 for all v ∈ V (G). By changing the sign of f ∗ (v) if necessary, we can ensure that hx, f ∗ (v)i ≥ ϑ(G)−1/2 for all v ∈ V (G), so that by Cauchy–Schwarz we obtain

* + ∗ ∗ n L(G) ≥ ||x|| f (v) ≥ x, f (v) ≥ . ∑ ∑ p ( ) v∈V (G) v∈V (G) ϑ G

Moreover, if G is vertex-transitive then Lovász [67] showed that ϑ(G)ϑ(G) = n, in which case the upper and lower bounds for L(G) coincide, so that we conclude q L(G) = nϑ(G).

In order to prove Theorem 4.1.3 for C2t+1-free graphs, we will need the following result proved implicitly by Erd˝os,Faudree, Rousseau, and Schelp [33]. It allows us to bound the chromatic number of the set of vertices at a fixed distance from a given vertex, for any graph without a cycle of prescribed length. Lemma 4.3.1. Let G be a graph having no cycle of length k and let i ≤ b(k − 1)/2c. Fix a vertex u0 in G and define Ai = {u ∈ V (G) : d(u, u0 ) = i} to be the set of vertices at a distance of exactly i from u0. Then the induced subgraph G[ Ai ] satisfies χ(G[ Ai ]) ≤ k − 2.

Proof. In the proof of Theorem 1 of [33], Erd˝os,Faudree, Rousseau, and Schelp show that if G does not contain a cycle of length k and i ≤ b(k − 1)/2c, then one can assign k − 2 labels to the vertices of Ai so that no two vertices having the same label are adjacent. Hence χ(G[ Ai ]) ≤ k − 2. 4.3 lovász ϑ-function for H-free graphs 49

Proof of Theorem 4.1.3. Let f be an orthonormal representation of G maximizing the largest eigenvalue of the corresponding Gram matrix M = M f . Let λ1 ≥ ... ≥ λn be the eigenvalues of M and observe that by Definition 1 of the ϑ- 2t+1 n 2t+1 function, ϑ G = λ1. Now note that tr( M ) = ∑i=1 λi and that λi ≥ 0 2t+1 n 2t+1 for all i since M is positive semidefinite. Thus we have λ1 ≤ ∑i=1 λi = 1/(2t+1) tr M2t+1 , and hence ϑ G ≤ tr M2t+1 . Therefore it will be enough for us to show that tr M2t+1 ≤ (6t)2t n. For convenience, given vertices u0, u1,..., uk, we define

k k−1 ! | | W(u0,..., uk) = ∏ h f (ui−1), f (ui)i = f (u0) ∏ f (ui) f (ui) f (uk) i=1 i=1 and note that W(u0, u1,..., u2t, u0) = 0 whenever u0u1 ... u2tu0 is not a closed walk in G, i.e. whenever one of the pairs u0u1,..., u2tu0 is a non-edge in G. Moreover, if u0u1 ... u2tu0 form a closed walk in G, then d(u0, ui) ≤ t for all i, so if we deﬁne t N (u0) = {v ∈ G : d(v, u0) ≤ t} to be the set of vertices at a distance of at most t from u0, we obtain

2t+1 tr M = ∑ W(u0, u1,..., u2t, u0) u0,u1,...,u2t∈V(G)

= ∑ ∑ W(u0, u1,..., u2t, u0). t u0∈V(G) u1,...,u2t∈N (u0) Thus if we deﬁne

Y(u0) := ∑ W(u0, u1,..., u2t, u0) t u1,...,u2t∈N (u0)

2t for u0 ∈ V(G), then it sufﬁces for us to show that Y(u0) ≤ (6t) for all u0, since we may then conclude

2t+1 2t tr M = ∑ Y(u0) ≤ (6t) n. u0∈V(G)

To bound Y(u0), we use Lemma 4.3.1. For any i ≤ t, deﬁne Ai = {u ∈ V(G) : d(u, u0) = i} to be the set of vertices at a distance of exactly i from u0. Since G has no cycle of length 2t + 1, we have by Lemma 4.3.1 that χ(G[Ai]) ≤ 2t, and so we let {B(i, 1),..., B(i, 2t)} be a partition of Ai into 2t independent sets. Note that for every closed walk u0 ... u2tu0, if we let di = d(u0, ui) denote the distance from u0 to ui, then |di+1 − di| ≤ 1. Thus we obtain

Y(u0) = ∑ ∑ ∑ W(u0, u1,..., u2t, u0). d ,...,d : a ,...,a ∈[2t] u1,...,u2t: 1 2t 1 2t ∈ ( ) d1=1, |di+1−di |≤1 ui B di,ai

Now since each B(i, j) is an independent set, it follows that { f (u) : u ∈ B(i, j)} | is an orthonormal set of vectors. Moreover, observe that Pi,j := ∑u∈B(i,j) f (u) f (u) 50 orthonormal representations of h-free graphs

is precisely the orthogonal projection onto the subspace spanned by { f (u) : u ∈ B(i, j)}. Thus for any d1,..., d2t such that d1 = 1, |di+1 − di| ≤ 1 for all i and for all a1,..., a2t ∈ [2t], we have

2t ! | | ∑ W(u0, u1,..., u2t, u0) = ∑ f (u0) ∏ f (ui) f (ui) f (u0) u1,...,u2t: u1,...,u2t: i=1 ui ∈B(di,ai ) ui ∈B(di,ai )   2t | | = f (u0) ∏ ∑ f (ui) f (ui)  f (u0) = u ∈B i 1 i di,ai 2t ! = f (u )| P f (u ) 0 ∏ di,ai 0 , i=1 and since any orthogonal projection P satisﬁes ||Pv|| ≤ ||v||, we may apply Cauchy– Schwarz to obtain 2t ! * 2t ! + f (u )| P f (u ) = f (u ) P f (u ) 0 ∏ di,ai 0 0 , ∏ di,ai 0 i=1 i=1 ! 2t ≤ || f (u )|| P f (u ) 0 ∏ di,ai 0 i=1

≤ || f (u0)|||| f (u0)|| = 1.

2t Since there are at most 3 sequences of integers (d1,..., d2t) such that d1 = 1 and |di+1 − di| ≤ 1 for all i, we therefore conclude

Y(u0) = ∑ ∑ ∑ W(u0, u1,..., u2t, u0) d ,...,d a ,...,a ∈[2t] u1,...,u2t : 1 2t 1 2t ∈ ( ) d1=1, |di+1−di |≤1 ui B di,ai ≤ ∑ ∑ 1 d1,...,d2t a1,...,a2t∈[2t] d1=1, |di+1−di |≤1 ≤ 32t(2t)2t = (6t)2t.

We now cite known constructions of Ct-free, K2,t-free, or Kt,(t−1)!+1-free graphs with many edges and optimal spectral gaps, in order to obtain Theorem 4.1.4. Note that some of the graphs described below have loops on some of their vertices, so to get a simple graph these loops should be removed. Since this only affects the adjacency matrix by subtracting a diagonal matrix with 1s and 0s on the diagonal, one can deduce from Weyl’s interlacing inequality that the eigenvalues only change by at most 1, not affecting the asymptotic bounds obtained below.

Proof of Theorem 4.1.4. For the C2t+1, C4, C6, C10, and Kt,(t−1)!+1-free graph constructions discussed below, see section 3 of the survey on pseudo-random graphs [63] by Krivelevich and Sudakov. 4.3 lovász ϑ-function for H-free graphs 51

As previously mentioned, Alon and Kahale [5] note that a modification of Alon’s construction [1] gives a graph with an optimal spectral gap which is, in particular, k C2t+1-free for any fixed t ≥ 1. Indeed, for any k such that 2 − 1 is not divisible by 4t + 3, the construction yields a 2k−1(2k−1 − 1)-regular graph G on n = 2(2t+1)k vertices which is C2t+1-free such that all eigenvalues of its adjacency matrix except the largest are bounded in absolute value by O(2k). The adjacency matrix A of k−1 k−1 such a graph therefore has largest eigenvalue λ1(A) = 2 (2 − 1) and all other eigenvalues bounded in absolute value by O(2k). Applying Definition 2 of ϑ G to the adjacency matrix of G, and using the fact that the smallest eigenvalue of G is negative (as the trace of the adjacency matrix is 0), we thus conclude

k−1 k−1 2 (2 − 1) 1/(2t+1) λ(n, C t+ ) ≥ ϑ G ≥ 1 + = Ω n . 2 1 O2k

The construction of a C4-free graph G with an optimal spectral gap and many edges comes from the projective space over a ﬁnite ﬁeld of order q = pα where p is a prime and α is an integer. It has n = (q3 − 1)/(q − 1) vertices, is (q2 − 1)/(q − 1)- √ regular, and all its eigenvalues beside the largest are in absolute value equal to q. Therefore, we obtain as above that

(q2 − 1)/(q − 1) − 1 λ(n, C ) ≥ ϑ G ≥ 1 + √ = Ω n1/4 . 4 q + 1

The C6-free graph G with an optimal spectral gap and many edges is the polarity graph of a generalized 4-gon. For q being an odd power of 2, G is a (q + 1)-regular graph with n = (q4 − 1)/(q − 1) vertices such that all eigenvalues besides the largest are bounded in absolute value by p2q. Thus we obtain

q + 1 1/6 λ(n, C6) ≥ ϑ G ≥ 1 + = Ω n . p2q

Similarly, the C10-free graph G with an optimal spectral gap and many edges is the polarity graph of a generalized 6-gon. For q being an odd power of 3, G is a (q + 1)-regular graph with n = (q6 − 1)/(q − 1) vertices such that all eigenvalues besides the largest are bounded in absolute value by p3q. Thus we obtain

q + 1 1/10 λ(n, C10) ≥ ϑ G ≥ 1 + = Ω n . p3q

The Kt,(t−1)!+1-free graph G with an optimal spectral gap and many edges is called a projective norm graph. For a prime p, G has pt − pt−1 vertices, is (pt−1 − 1)-regular, and all eigenvalues besides the largest are in absolute value at most p(t−1)/2. Thus we obtain

t−1 p − 1 1 (1−1/t) λ(n, K ( − ) + ) ≥ ϑ G ≥ 1 + = Ω n 2 . t, t 1 ! 1 p(t−1)/2 52 orthonormal representations of h-free graphs

The following construction of a K2,t+1-free graph with many edges is due to Füredi [38]. As he did not show that this construction has an optimal spectral gap, we prove it below. Let q be a prime power such that t divides q − 1 and let F be a finite field of order q. Let h ∈ F be an element of order t and let H = {1, h, h2,..., ht−1}. Define the equivalence relation on F × F \ {(0, 0)} by (a, b) ∼ (a0, b0) iff there exists c ∈ H such that (a0, b0) = c · (a, b). Let ha, bi denote the equivalence class of (a, b) under the relation ∼. Now define G to be the graph whose vertices are the equivalence classes (F × F \ {(0, 0)}) / ∼ such that there is an edge between ha, bi and ha0, b0i iff aa0 + bb0 ∈ H. Each equivalence class has t elements, and therefore G has n = (q2 − 1)/t vertices. Moreover, for each vertex (a, b) ∈ F × F \ {(0, 0)}, there are q solutions (x, y) to the equation ax + by = c for any c ∈ H, and therefore ha, bi has degree t2q/t2 = q. Now let ha, bi, ha0, b0i be a pair of distinct vertices and consider their common neighborhood. To determine its size, we must determine the number of solutions (x, y) to the equations

ax + by = d a0x + b0y = e

where d, e ∈ H. If there exists c such that a0 = ca, b0 = cb, then the equations have no solutions, since otherwise we would have e = cax + cby = cd, which would imply that c ∈ H, contradicting the fact that ha, bi 6= ha0, b0i. Thus ha, bi and ha0, b0i have no common neighbors in this case. Otherwise if there does not exist c such ! a b that a0 = ca, b0 = cb, then the matrix is invertible and hence the system a0 b0 of equations has exactly one solution (x, y) for each choice of d, e ∈ H. As there are t2 choices for d and e, we obtain a total of t2 solutions, which implies that there are t2/t = t vertices in the common neighborhood of ha, bi and ha0, b0i. Thus G has no copy of K2,t+1. Now let A be the adjacency matrix of G, indexed by the vertices ha, bi, and consider A2. Since G is q-regular, the diagonal entries of A2 will all be q. The off- 2 h i h 0 0i diagonal entry Aha,bi,ha0,b0i is the number of common neighbors of a, b and a , b , which by the previous discussion is either 0 or t depending on whether or not there exists c such that a0 = ca, b0 = cb. Thus if we let Q be the {0, 1} matrix indexed 0 0 by the vertices of G so that Qha,bi,ha0,b0i = 1 iff ha, bi and ha , b i have no common neighbors, then we have A2 = (q − t)I + tJ − tQ where I is the identity matrix and J is the all-ones matrix. Now for any given ha, bi, observe that we must have c ∈ (F\{0}) \ H in order for a0 = ca, b0 = cb to yield (a0, b0) 6= (0, 0) such that ha0, b0i 6= ha, bi. This gives q − 1 − t choices for c and therefore there are exactly (q − 1 − t)/t many vertices ha0, b0i that have no common neighbors with ha, bi, so that Q is a matrix with (q − 1 − t)/t ones in each row. By the Perron–Frobenius theorem, the largest eigenvalue of Q is λ1(Q) = (q − 1 − t)/t 4.3 lovász ϑ-function for H-free graphs 53

| with eigenvector (1, . . . , 1) , and all other eigenvalues satisfy |λi(Q)| ≤ λ1(Q). | Since J has largest eigenvalue λ1(J) = n also with the eigenvector (1, . . . , 1) and 2 all other eigenvalues of J are 0, we conclude that for all i ≥ 2, λi(A ) = q − t − tλn−i+1(Q) and hence q − 1 − t |λ (A2)| ≤ q − t + t · = 2q − 2t − 1. i t Now since G is q-regular, again by the Perron–Frobenius theorem we have that the | largest eigenvalue of A is λ1(A) = q with eigenvector (1, . . . , 1) . For all i ≥ 2, it 2 2 therefore follows that there exists j ≥ 2 such that λi(A) = λj(A ), and thus we conclude p |λi(A)| ≤ 2q − 2t − 1. Finally, applying Deﬁnition 2 of ϑ G with the matrix A, we obtain

λ1(A) q 1/4 1/4 λ(n, K2,t+1) ≥ ϑ G ≥ 1 − ≥ 1 + p = Ω t n . λn(A) 2q − 2t − 1

We now give a proof of Theorem 4.1.5, using an approach similar to that which 1−2/t was used by Alon and Kahale to prove λ(n, Kt) ≤ O(n ) in [5].

Proof of Theorem 4.1.5. We proceed by induction on n. For n = 1 the claim holds trivially. Now suppose n ≥ 2 and let G be an H-free graph on n vertices. Deﬁne U = {v ∈ V(G) : d(v) ≤ 4cnα} and W = V(G)\U. It follows from Deﬁnition 4 of the ϑ-function that ϑG ≤ ϑ G[U] + ϑ G[W] . Moreover, observe that

4cnα|W| ≤ ∑ d(v) ≤ 2 ex(n, H) ≤ 2cn1+α, v∈W so |W| ≤ n/2, and hence by the induction hypothesis √ ch n α/2 ϑ G[W] ≤ λ(n/2, H) ≤ 20 . α 2 Thus it remains to bound ϑ G[U] . To this end let f be an orthonormal representation of G[U] maximizing the largest eigenvalue λ1(M) of the corresponding Gram matrix M = M f . By Definition 1 of the ϑ-function, we have ϑ G[U] = 0 λ1(M). Now fix u ∈ U and define N (u) = {w ∈ U : uw ∈ E(G)} to be the neighborhood of u in G[U]. Since G[U] has no copy of H, we have by the same argument as in the proof of Theorem 4.1.11 that N0(u) can be partitioned into at most h in- 2 dependent sets, so that by Parseval’s inequality, ∑w∈N0(u) h f (u), f (w)i ≤ h. Since |N0(u)| ≤ d(u) ≤ 4cnα, we conclude via Cauchy–Schwarz that √ ∑ | h f (u), f (w)i | ≤ 4cnαh. w∈N0(u) 54 orthonormal representations of h-free graphs

Putting everything together, we have √ √ ch n α/2 ϑG ≤ ϑ G[U] + ϑ G[W] ≤ 3 cnαh + 20 α 2 ! √ 20 1 α/2 = cnαh 3 + . α 2

Now to complete the proof, we use the fact that e−x ≤ 1 − x/2 for 0 ≤ x ≤ 1 to conclude 20 1 α/2 20 ln(2)α 20 3 + ≤ 3 + 1 − ≤ . α 2 α 4 α Corollary 4.1.6 will now follow from known upper bounds on Turán numbers.

Proof of Corollary 4.1.6. Recently, Bukh and Tait [18] showed that ex(n, θt,s) ≤ O ts1−1/tn1+1/t , generalizing the well-known upper bounds 1+1/t ex (n, C2t) ≤ O tn due to Bondy and Simonovits [15], and ex (n, K2,t) ≤ 1+1/t O tn due to Füredi [38]. Since θt,s consists of a tree together with an additional vertex, we may apply Theorem 4.1.5 to obtain the desired upper bounds on λ(n, H).

Remark 4.3.2. For even cycles, Bukh and Jiang [17] recently improved the bound on √ 1+1/t the Turán number to ex(n, C2t) ≤ O t log t n for n sufﬁciently large rela- 7/4p 1/(2t) tive to t, which would imply λ(n, C2t) ≤ O t log t n via Theorem 4.1.5. Nonetheless, in the appendix we show how to obtain the better bound λ(n, C2t) ≤ O t n1/(2t) via a different argument. Theorem 4.1.8 will follow from an argument similar to that of Theorem 4.1.5, except that we will have to replace the result that the chromatic number of a neighborhood is bounded, with a bound on the ϑ-function of a neighborhood which will be obtained inductively.

Proof of Theorem 4.1.8. We proceed by induction on n and t, where ct,s will be deﬁned recursively. For s ≥ t = 2, let c2,s be the constant such that λ(n, K2,s) ≤ 4.3 lovász ϑ-function for H-free graphs 55

3/4 1/4 c2,ss n as given by Corollary 4.1.6. Now suppose s ≥ t ≥ 3. For n = 1, the claim trivially holds for ct,s ≥ 1. Now let n ≥ 2. Kövari, Sós, and Turán [61] showed that there exists a constant at,s such that 2−1/t ex(n, Kt,s) ≤ at,sn . As in the proof of Theorem 4.1.5, deﬁne U = {v ∈ V(G) : 1−1/t d(v) ≤ 4at,sn }, W = V(G)\U, and observe that by Deﬁnition 4 of the ϑ- function, ϑG ≤ ϑ G[U] + ϑ G[W] . Moreover, observe that

1−1/t 2−1/t 4at,sn |W| ≤ ∑ d(v) ≤ 2 ex(n, Kt,s) ≤ 2at,sn , v∈W so |W| ≤ n/2, and hence by the induction hypothesis

n 1−2/t+1/(t 2t−1) ϑ G[W] ≤ λ(n/2, K ) ≤ c . s,t t,s 2 To bound ϑ G[U] , let f be an orthonormal representation of G[U] maximizing the largest eigenvalue λ1(M) of the corresponding Gram matrix M = M f , so that 0 we have ϑ G[U] = λ1(M). Now ﬁx u ∈ U and let N (u) = {w ∈ U : uw ∈ E(G)} be the neighborhood of u in G[U]. Note that G[U] has no copy of Kt−1,s, so that via Deﬁnition 4 of the ϑ-function and induction, we have

2 0 ∑ h f (u), f (w)i ≤ ϑ G[U] ≤ λ(|N (u)|, Kt−1,s) w∈N0(u) 0 1−2/(t−1)+1/((t−1) 2t−2) ≤ ct−1,s |N (u)| .

0 1−1/t Thus using the fact that |N (u)| ≤ 4at,sn and applying Cauchy–Schwarz, we conclude q 0 0 1−2/(t−1)+1/((t−1) 2t−2) ∑ | h f (u), f (w)i | ≤ |N (u)|ct−1,s|N (u)| w∈N0(u)

t−1 √ 1−1/(t−1)+1/((t−1) 2 ) 1−2/t+1/(t 2t−1) ≤ 4 ct−1,sat,s n . As in the proof of Theorem 4.1.5, we therefore obtain ϑ G[U] ≤ max | h f (u), f (w)i | ∈ ∑ u U w∈U   = max1 + | h f (u), f (w)i | u∈U ∑ w∈N0(u)

t−1 √ 1−1/(t−1)+1/((t−1) 2 ) 1−2/t+1/(t 2t−1) ≤ 1 + 4 ct−1,sat,s n .

Thus if we set √ 1−1/(t−1)+1/((t−1) 2t−1) 1 + ct−1,sat,s ct,s = − , 1 − (1/2)1−2/t+1/(t 2t 1) 56 orthonormal representations of h-free graphs

then we conclude the desired result

1−2/t+1/(t 2t−1) ϑ G ≤ ϑ G[U] + ϑ G[W] ≤ ct,s n .

4.4 concluding remarks

We have seen that for H ∈ {C2t+1, C4, C6, C10, K2,t} ﬁxed and n large, Theorem 4.1.4 and Corollary 4.1.6 provide bounds on λ(n, H) that are asymptotically tight. How- ever, the lower bound in Theorem 4.1.4 for λ(n, Kt,s) with s ≥ t ≥ 3 does not match the upper bound obtained in Theorem 4.1.8, so determining the correct asymptotic dependence on n is an interesting problem. Indeed, for n t → ∞, we have

1/2 − o(1) ≤ logn λ(n, Kt,s) ≤ 1 − o(1), where the lower bound is coming from graphs with optimal spectral gaps that are almost extremal for the Turán number, so that we cannot hope to do better with such constructions. On the other hand, we know

1 − o(1) ≤ logn λ(n, Kt) ≤ 1 − o(1), for n t → ∞, and it would therefore be interesting to determine if the asymptotic behavior of λ(n, H) is different for H = Kt versus H = Kt,s. For H = K2,t, even though we know the asymptotic behavior of λ(n, H), we are only able to show that

1/4 1/4 3/4 1/4 Ω t n ≤ λ(n, K2,t) ≤ O t n ,

so it would be interesting to determine the correct dependence of λ(n, K2,t) on t. Acknowledgments. We thank Shoham Letzter and Benny Sudakov for working with the author in order to obtain these results. We would also like to thank Boris Bukh and Chris Cox for stimulating discussions. 5 THEMIN-RANKOFRANDOMGRAPHS

5.1 introduction

In this chapter we discuss the notion of the min-rank of a graph over a field, defined as follows. Definition 5.1.1. The min-rank of a graph G on the vertex set [n] = {1, 2, . . . , n} over a field F, denoted by mrF(G), is the minimum possible rank of a matrix n×n M ∈ F with nonzero diagonal entries such that Mi,j = 0 whenever i and j are distinct nonadjacent vertices of G. Recall that the notion of the min-rank over the real field R, with the added requirement that the representing matrix M be positive semidefinite, gives rise to the minimum semidefinite rank msr(·) studied in Chapter 4. Thus for any graph G, we have mrR(G) ≤ msr(G). We also note that the min-rank over finite fields has been studied for its connections to the Shannon capacity [47] and to linear index coding [10]. For fixed p ∈ (0, 1) and large n, Knuth [58] raised the problem of determining the typical value of msr(G(n, p)) where G(n, p) is the random graph on n vertices with each√ edge chosen independently with probability p, mentioning that it is at least Ω( n). The analogous problem for the min-rank over finite fields was raised by Lubetzky and Stav [70] also in the context√ of linear index coding. Haviv and Langberg [50] proved a lower bound of Ω( n) for the min-rank of G(n, p) over any fixed finite field and for any constant p. In a recent beautiful paper, Golovnev, Regev, and Weinstein [43] substantially improved the aformentioned results by showing that for any finite field F = F(n) and every p = p(n) ∈ (0, 1), the min-rank of the random graph G(n, p) over F is with high probability at least n log(1/p) Ω . log(n|F|/p) Since the min-rank of every graph over any field is at most the chromatic number of its complement, the known results about the behavior of the chromatic number of random graphs show that the above estimate is tight up to a constant factor for every finite field of size at most nO(1) and for every p which is not too close to 0 or 1, e.g., for all n−1 ≤ p ≤ 1 − n−0.99. This, however, provides no information for infinite fields, and in particular for the real field. Our main result is an extension of this result to all finite or infinite fields. Here and in what follows, the expression “with high probability” (w.h.p. for short) means “with probability tending to 1 as n goes to infinity”.

57 58 the min-rank of random graphs

Theorem 5.1.2. Let F = F(n) be a field and assume that p = p(n) satisfies n−1 ≤ p ≤ 1. Then w.h.p. n log(1/p) mrF(G(n, p)) ≥ . 80 log n The proof combines the method of Golovnev, Regev, and Weinstein with tools from linear algebra, most notably an estimate of Rónyai, Babai, and Ganapathy [78] for the number of zero patterns of a sequence of multivariate polynomials over a field. The result for the real field settles the problem of Knuth mentioned above. We conclude this introduction by making several remarks. Remark 5.1.3 (Tightness of Theorem 5.1.2). Theorem 5.1.2 is tight up to the value 1 −1 −0.99 of the multiplicative constant 80 for every field F and every n ≤ p ≤ 1 − n . 1 Indeed, for every graph G and every field F we have mrF(G) ≤ χ(G) , and it is known that for p in the above range, G = G(n, p) w.h.p. satisfies χ(G) = ( n log(1/p) ) Θ log n , see [14, 53]. This proves the following result: Theorem 5.1.4. Let F = F(n) be a field and assume that p = p(n) satisfies n−1 ≤ p ≤ 1 − n−0.99. Then w.h.p. n log(1/p) mrF(G(n, p)) = Θ . log n Remark 5.1.5 (Amplifying the success probability in Theorem 5.1.2). The proof of Theorem 5.1.2 gives a bound of n−Ω(1) on the probability that G = G(n, p) (for ≥ −1 ( ) < n log(1/p) p n ) satisfies mrF G 80 log n . Using Azuma’s inequality for the vertex exposure martingale, one can show that mrF(G) is highly concentrated around its ( n log(1/p) ) expectation, which is Ω log n by Theorem 5.1.2. This way, one can deduce that ( ) ≥ ( n log(1/p) ) − −Ω(n/ log2 n) mrF G Ω log n holds with probability at least 1 e , see [43] for a detailed argument. The rest of this chapter is organized as follows. The proof of Theorem 5.1.2 is given in Section 5.2 and Section 5.3. Section 5.4 contains a number of applications of our main theorem to the study of various geometric representations of random graphs. Section 5.5 contains some concluding remarks and open problems.

5.2 preliminaries

Deﬁnition 5.2.1. An (n, k, s)-matrix (over some ﬁeld F) is an n × n matrix M of rank

k with s nonzero entries and containing rows Ri1 ,..., Rik and columns Cj1 ,..., Cjk

such that Ri1 ,..., Rik is a row basis for M, Cj1 ,..., Cjk is a column basis for M, and

the overall number of nonzero entries in all 2k vectors Ri1 ,..., Rik , Cj1 ,..., Cjk is at most 4ks/n.

1 Given a proper coloring of G, deﬁne M by Mi,j = 0 if i, j lie in different color classes, and Mi,j = 1 otherwise. It is easy to see that the rank of M is the number of colors, and that Mi,j = 0 whenever (i, j) ∈/ E(G). 5.2 preliminaries 59

The following is the key lemma in [43]. Lemma 5.2.2. Let F be any field and let M ∈ Fn×n be a matrix of rank k. Then there exist integers n0, k0, and s0 with k0/n0 ≤ k/n such that M contains an (n0, k0, s0)-principal- submatrix (that is, a principal submatrix that is an (n0, k0, s0)-matrix). m The zero-pattern of a sequence (y1,..., ym) ∈ F is the sequence (z1,..., zm) ∈ m {0, ∗} such that zi = 0 if yi = 0 and zi = ∗ if yi 6= 0. For a sequence of polynomials ¯ ¯ f = ( f1,..., fm) over a field F in variables X1,..., XN, the set of zero-patterns of f is the set of all zero-patterns of sequences obtained by assigning values from F to the variables X1,..., XN in ( f1,..., fm). We define the zero-pattern of a matrix M ∈ Fn×n, and the set of zero-patters of a matrix whose entries are polynomials, by treating the matrix as a sequence of length n2. Rónyai, Babai, and Ganapathy [78] gave the following bound for the number of zero-patterns of a sequence of polynomials: ¯ Lemma 5.2.3. Let f = ( f1,..., fm) be a sequence of polynomials in N variables over a ¯ md+N field F, each of degree at most d. Then the number of zero-patterns of f is at most ( N ). We now state and prove the key lemma of this paper.

n 2 20ks/n Lemma 5.2.4. The number of zero-patterns of (n, k, s)-matrices is at most (k) · n . Proof. It is easy to see that the lemma holds for k ≥ n − 1 (since the number of n2 2s zero-patterns of n × n matrices with s nonzero entries is clearly at most ( s ) ≤ n ), n 2 so we may assume for convenience that k ≤ n − 2. The term (k) corresponds to the number of ways to choose the sequences (i1,..., ik) and (j1,..., jk) from Deﬁnition 5.2.1 (that is, the number of ways to choose the positions of the rows

Ri1 ,..., Rik and the columns Cj1 ,..., Cjk ). From now on we assume without loss of generality that (i1,..., ik) = (j1,..., jk) = (1, . . . , k). The number of ways to choose a set F ⊆ ([k] × [n]) ∪ ([n] × [k]) of at most 4ks/n entries which are allowed to be nonzero is at most !2ks/n 4ks/n 2kn e · n4 ≤ ≤ 8ks/n ∑ 2 n , t=0 t s where the ﬁrst inequality follows from the fact that for all 0 < x < 1, we have

4ks/n 2kn 4ks/n 2kn ∑ x4ks/n ≤ ∑ xt ≤ (1 + x)2kn ≤ ex2kn t=0 t t=0 t and setting x = s/n2. So it is enough to show that for every fixed F ⊆ ([k] × [n]) ∪ ([n] × [k]) as above, there are at most n12ks/n zero-patterns of matrices for which the first k rows form a row basis, the first k columns form a column basis, and for every (i, j) with min(i, j) ≤ k and (i, j) ∈/ F, the (i, j)-entry is zero. Let M = (Mi,j) be such a matrix, and denote by M0 the submatrix of M on [k] × [k]. We claim that M0 is invertible. Indeed, let M00 be the submatrix consisting of the first k rows of 60 the min-rank of random graphs

M. Then rank(M00) = k (because the rows of M00 form a row basis of M) and the columns of M0 span the column space of M00 (because the first k columns of M span its column space). It follows that the columns of M0 are linearly independent, as required. Fix any k + 1 ≤ ` ≤ n. The `-th column of M is a linear combination of the first k columns of M. The coefficients in this linear combination are the coordinates of the unique solution to the system   M1,` 0  .  M · x =  .  .  .  Mk,` By Cramer’s rule, this solution can be expressed as

 (`) 0  f1,`(y )/ det(M )  .   .  ,  .  (`) 0 fk,`(y )/ det(M )

where f1,`,..., fk,` are polynomials of degree k (which do not depend on the matrix (`) k k M), and the vector of variables y contains the entries (Mi,j)i,j=1 and (Mi,`)i=1. We see that for every k + 1 ≤ ` ≤ n and 1 ≤ i ≤ n, we have

1 k M = f (y(`)) · M . i,` ( 0) ∑ j,` i,j det M j=1

This means that every entry of M can be given as a polynomial of degree k + 1 0 in the entries (Mi,j)min(i,j)≤k, divided by the nonzero polynomial det(M ). Since Mi,j = 0 if min(i, j) ≤ k and (i, j) ∈/ F, it is enough to take (Mi,j)(i,j)∈F as the sequence of variables of all polynomials. We conclude that the zero-pattern of M is the zero-pattern of a sequence of n2 polynomials in |F| ≤ 4ks/n variables, each + 1 of degree (at most) k 1 (note that removing the factor det(M0) does not change the zero-pattern of M, and that all polynomials are independent of M). By Lemma 5.2.3, the number of zero-patterns of this matrix of polynomials is at most

(k + 1)n2 + 4ks/n (k + 2)n2 ≤ ≤ (k + 2)4ks/nn8ks/n ≤ n12ks/n, 4ks/n 4ks/n as required.

Finally, we will need the following simple lemma from [43] (which follows, with a slightly better constant, from Turán’s Theorem). Lemma 5.2.5. Every n × n matrix of rank k having nonzero entries on the main diagonal contains at least n2/(4k) nonzero entries. 5.3 the min-rank of random graphs 61

5.3 themin-rank of random graphs

≥ − −1 n log(1/p) = ( ) Proof of Theorem 5.1.2. If, say, p 1 n , then log n o 1 , so the theorem holds trivially. So from now on we assume that p < 1 − n−1. Suppose that G is an n-vertex graph with mrF(G) ≤ k. Then by definition, there is a n × n matrix M over F of rank at most k, such that all entries of M on the main diagonal are nonzero, and such that Mi,j = 0 whenever (i, j) ∈/ E(G). By Lemma 5.2.2, M contains an (n0, k0, s0)-principal submatrix M0 with k0/n0 ≤ k/n. We conclude that for every graph G satisfying mrF(G) ≤ k, there is a set U ⊆ V(G) and an (n0, k0, s0)-matrix M0, where n0 = |U| and k0/n0 ≤ k/n, such that all entries of M0 on the main diagonal are nonzero, and such that for every pair 0 0 0 0 of distinct i, j ∈ U, we have Mi,j = 0 whenever (i, j) ∈/ E(G). For given n , k , s , n 0 0 0 the number of choices for U is (n0), and the number of zero-patterns of (n , k , s )- n0 2 020k0s0/n0 matrices is at most (k0 ) · n by Lemma 5.2.4. Fixing U and the zero-pattern of M0, the probability that G = G(n, p) satisfies the above event with respect to 0 0 U, M0 is at most p(s −n )/2, since there are at least (s0 − n0)/2 pairs 1 ≤ i < j ≤ n0 0 0 for which either Mi,j 6= 0 or Mj,i 6= 0, and each such pair must span an edge in G. By Lemma 5.2.5 we have s0 ≥ n02/(4k0) ≥ n0n/(4k). Hence, the probability that G = G(n, p) satisfies mrF(G) ≤ k is at most

0 n n k/n 02 0 0 0 n n 020k s /n (s0−n0)/2 ∑ ∑ ∑ 0 · 0 · n · p 0= 0 0 0 n n k n 1 k =1 s ≥n · 4k n n0k/n 0 0 0 0 s ≤ ∑ ∑ nn +2k · p−n /2 ∑ n20k/n p1/2 (5.1) 0= 0 0 0 n n 1 k =1 s ≥n · 4k

≤ n log(1/p) 20k/n ≤ ( )1/4 For k 80 log n we get n 1/p , and so

s0 20k/n 1/2 s0/4 n0· n 1 −5n0 log n 1 n p ≤ p ≤ p 16k · ≤ e · ∑ ∑ 1/4 1/4 0 0 n 0 0 n 1 − p 1 − p s ≥n · 4k s ≥n · 4k 0 1 = n−5n · . 1 − p1/4

Hence, (5.1) is at most

n n0k/n n 1 0 0 0 0 1 0 0 · nn +2k · p−n · n−5n ≤ · n4n · n−5n 1/4 ∑ ∑ 1/4 ∑ 1 − p n0=1 k0=1 1 − p n0=1 n 1 0 = · n−n . 1/4 ∑ 1 − p n0=1 If (say) p ≤ 1/2, then the above sum is clearly o(1). In the complementary case p > 1/2 we have k = O(n/ log n), and so in (5.1) we can restrict ourselves to n0 62 the min-rank of random graphs

satisfying n0 ≥ n/k = Ω(log n) (as otherwise there are no k0 between 1 and n0k/n). Now, recalling the assumption p < 1 − n−1, we see that (5.1) evaluates to o(1). This completes the proof.

5.4 geometric representations of random graphs

Orthonormal representations

In 1994, Knuth [58] asked what the typical value of minimum semidefinite rank of G(n, p) is for fixed p. As noted in Section 5.1, the min-rank over the real field R is related to the minimum semidefinite rank studied in Chapter 4. Indeed, for any graph G with vertex set [n] and any orthonormal representation f of G, the corresponding Gram matrix M = M f satisfies Mi,i = h f (i), f (i)i = 1 for all i and Mi,j = h f (i), f (j)i = 0 whenever i and j are not adjacent. Thus M represents G, and we therefore conclude mrR(G) ≤ msr(G). ( ( )) ≥ ( n log(1/p) ) Hence, Theorem 5.1.2 shows that msr G n, p Ω log n with high probability. On the other hand, in Chapter 4 Section 4.1 we note that msr(G) ≤ χ(G). Thus, by the same argument as in Remark 5.1.3, we conclude that our bound is tight for n−1 ≤ p ≤ 1 − n−0.99. This proves the following theorem, settling Knuth’s problem. Theorem 5.4.1. For every p = p(n) satisfying n−1 ≤ p ≤ 1 − n−0.99, we have with high probability that n log(1/p) msr(G(n, p)) = Θ . log n

Unit distance graphs

A complete unit-distance graph in Rd is a graph whose set of vertices is a ﬁnite subset of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A unit distance graph in Rd is any subgraph of such a graph. Unit distance graphs have been considered in several papers, see, e.g., [6] and the references therein. Note that if u, v ∈ Rd 2 are two adjacent vertices of such a unit distance graph, then ku − vk2 = 1. Let d d u1, u2,..., un ∈ R be the vertices of a unit distance graph G in R . Then the n × n 2 matrix M deﬁned by Mi,j = 1 − kui − ujk2 is a real matrix in which every entry on the diagonal is 1 and for every pair of distinct adjacent vertices ui, uj, Mi,j = 0. This implies that the rank of the matrix M must be at least mrR(G). On the other hand, it is easy to see that the rank of M is at most d + 2. Indeed M can be expressed 2 2 as a sum of three matrices A, B, C where Ai,j = 1 − kuik , Bi,j = −kujk and t Ci,j = 2ui vj. As all columns of A and all rows of B are identical, A and B are of 5.4 geometric representations of random graphs 63 rank 1. The matrix C is twice the Gram matrix of vectors in Rd, and hence its rank is at most d. Therefore M has rank at most d + 2. It is also clear that every graph of chromatic number d is a unit distance graph in Rd−1. Indeed, the d vertices d−1 x1,..., xd of a regular simplex of diameter 1 in R can be used to represent all vertices of G, assigning xi to all vertices in color class number i of G, for 1 ≤ i ≤ d. This establishes the following result. Theorem 5.4.2. For every p = p(n) satisfying n−0.99 ≤ p ≤ 1 − n−1, the minimum dimension d such that the random graph G = G(n, p) is a unit distance graph in Rd is, n log(1/(1−p)) w.h.p., Θ log n .

Graphs of touching spheres

The notion of a unit distance graph can be extended as follows. Call a graph G on d n vertices a graph of touching spheres in R if there are spheres S1, S2 ..., Sn in d R , where the sphere Si is centered at ui and its radius is ri, and for every pair of adjacent vertices i and j, the two corresponding spheres Si and Sj touch each other and their convex hulls have disjoint interiors. That is, the distance between ui and uj is exactly ri + rj. Note that if ri = 1/2 for all i, then this is exactly the deﬁnition of a unit distance graph. For ui and ri as above, the matrix M = (Mi,j) 2 2 where Mi,j = (ri + rj) − kui − ujk has nonzero diagonal elements and Mi,j = 0 for every pair of adjacent vertices i, j. Furthermore, M can be written as a sum of 2 2 2 2 t the four matrices in which the (i, j)-th entry is ri − kuik , rj − kujk , rirj, and 2ui uj, respectively. These have ranks at most 1, 1, 1, and d, respectively, showing that the rank of any such matrix M is at most d + 3. The chromatic number of G provides a representation as before (even as a unit distance graph), implying the following extension of Theorem 5.4.2. Theorem 5.4.3. For every p = p(n) satisfying n−0.99 ≤ p ≤ 1 − n−1, the minimum dimension d such that the random graph G = G(n, p) is a graph of touching spheres in Rd n log(1/(1−p)) is, w.h.p., Θ log n .

Graphs deﬁned by a polynomial

Let P = P(x, y) = P(x1, x2,..., xd, y1, y2,..., yd), where x = (x1,... xd) and y = (y1,..., yd), be a polynomial of 2d variables over a ﬁeld F, and assume that it satis- ﬁes P(x, y) = P(y, x) for all x, y ∈ Fd. Say that a graph G on n vertices 1, 2, . . . , n is a P-graph over Fd if there are vectors x(1),..., x(n) ∈ Fd such that P(x(i), x(i)) 6= 0 for all 1 ≤ i ≤ n, and for every pair of distinct adjacent vertices i, j, P(x(i), x(j)) = 0. Thus, for example, unit distance graphs correspond to the polynomial 1 − kx − yk2. We will often think of P as a sequence of polynomials, indexed by n (so the number of variables is allowed to grow with n). 64 the min-rank of random graphs

(i) (j) For any P-graph as above, the matrix M given by Mi,j = P(x , x ) vanishes in every entry corresponding to adjacent vertices, and has nonzero entries on the main diagonal. If the degree of P is large, then even a small number of variables 2d can be enough to represent all n-vertex graphs as P-graphs. Indeed, if for example, d d the field is F2, d = log2 n, and P = ∏i=1(1 + xi + yi), then for the set X = {0, 1} of n = 2d vertices, we have P(x, x0) 6= 0 if and only if x = x0, meaning that every graph on n vertices is a P-graph, although the number of variables is only O(log n). On the other hand, if P is of degree at most 3, it is not difficult to see that if G is a P-graph then the rank of the matrix M defined as above is at most 2d + 1. To see this, write P in the form

d d P = c + ∑ xi fi(y) + ∑ yjhj(x). i=1 j=1

Next deﬁne, for each vector x = (x1, x2,..., xd), two vectors F(x) and H(x) of length 2d + 1 each, as follows:

F(x) = (1, x1, x2,..., xd, h1(x), h2(x),..., hd(x)),

H(x) = (c, f1(x), f2(x),..., fd(x), x1, x2,..., xd).

Thus for every x = (x1,..., xd) and y = (y1,..., yd), P(x, y) is exactly the inner product of F(x) with H(y). This shows that if G is a P-graph then the matrix M above can be written as a product of the n × (2d + 1) matrix whose rows are the vectors F(x) by the (2d + 1) × n matrix whose columns are the vectors H(x) (where in both cases x goes over all vectors representing the vertices of G). This shows that indeed the rank of M is at most 2d + 1 and implies the following.

Theorem 5.4.4. Let P = P(x1, x2,..., xd, y1, y2,..., yd) be a polynomial of degree at most 3 over a ﬁeld F and let p = p(n) satisfy n−1 ≤ p ≤ 1. If the random graph = ( ) ( ) n log(1/(1−p)) G G n, p is a P-graph with probability Ω 1 , then d is at least Ω log n . Note that the proof above works for every polynomial P with O(d) monomials like, for example, d 4 4 P(x, y) = 1 − kx − yk4 = 1 − ∑(xi − yi) , i=1 or even every polynomial P which is the sum of O(d) terms, each being either a product of a monomial in the variables of x times any function of those in y, or vice versa.

Spaces of polynomials

For a ﬁeld F and a linear space S of polynomials in F[x1, x2 ..., xm], a graph G on the vertices 1, 2, . . . , n has a representation over S if for every vertex i there are m Pi ∈ S and vi ∈ F so that Pi(vi) 6= 0 for all i and Pi(vj) = 0 for every two distinct 5.5 concluding remarks 65 nonadjacent vertices i and j. As shown in [2], if G has such a representation then its Shannon capacity is at most the dimension of the linear space S. It is easy to see that the rank of the matrix M = (Mi,j) = (Pi(vj)) is at most the dimension of S. Therefore we get the following.

Theorem 5.4.5. Let S be a linear space of polynomials in variables x1, x2,..., xm over a ﬁeld F and let p = p(n) satisfy n−1 ≤ p ≤ 1. If G = G(n, p) has a representation over S ( ) S n log(1/p) with probability Ω 1 , then n is at least Ω log n .

5.5 concluding remarks

• We have shown that for all n−1 ≤ p ≤ 1 − n−0.99 and for any finite or infinite field F, the min-rank of the random graph G(n, p) over F satisfies, w.h.p., ( ) = ( n log(1/p) ) = −1 ( ) mrF G Θ log n . For p n this gives a lower bound of Ω n , and as n is always a trivial upper bound and the function mrF(G) for G = G(n, p) is clearly monotone decreasing in p, it follows that for all 0 ≤ p ≤ n−1, −1 mrF(G) = Θ(n). In the other extreme, for p ≥ 1 − O(n ), the graph G = G(n, p) satisfies w.h.p. χ(G) = Θ(1), and hence mrF(G) = Θ(1). So the only regime in which there is a gap of more than a constant factor between the lower bound of Theorem 5.1.2 and the typical value of χ(G(n, 1 − p)) is when ω(n−1) ≤ 1 − p ≤ n−1+o(1). In all the range of p discussed in this paper, the min-rank of a graph is equal, up to a constant factor, to the chromatic number of its complement. It will be interesting to decide how close these two quantities really are, and in particular, to decide whether or not for G = G(n, 1/2),

n mrR(G) = (1 + o(1))χ(G) = (1 + o(1)) . 2 log2 n

• It was shown in [47] that the min-rank of a graph over any field is an upper bound for its Shannon capacity. In particular, the infimum of mrF(G) over all fields F is such an upper bound. Combining our technique here with a recent result of Nelson (Theorem 2.1 in [72]) that extends the one of [78], we can show that for the random√ graph G(n, 1/2) this bound is weaker than the theta function, which is Θ( n) [56]. More generally, we have the following. Theorem 5.5.1. For every n−1 ≤ p ≤ 1, the random graph G = G(n, p) satisfies ( ) ≥ n log(1/p) F w.h.p. that mrF G Ω log n for every field . It is worth noting that it follows from results of Grosu [44] and of Tao [82] that the min-rank of a graph G over C is a lower bound for its min-rank over every field F whose characteristic is sufficiently large as a function of G. By combining these results with Theorem 5.1.2, we immediately get that for −1 every n ≤ p ≤ 1, the random graph G = G(n, p) w.h.p. satisfies mrF(G) ≥ 66 the min-rank of random graphs

n log(1/p) F Ω log n for every ﬁeld of characteristic which is sufﬁciently large as a function of n. The stronger assertion of Theorem 5.5.1 follows by replacing the result of [78] (Lemma 5.2.3) by that of [72] in the proof of Theorem 5.1.2.

• In general, the min-rank of a graph may depend heavily on the choice of the ﬁeld. To see this we use the well-known fact that for any graph G on n vertices and for any ﬁeld F,

mrF(G) · mrF(G) ≥ n.

Indeed, if A = (Ai,j) and B = (Bi,j) are representations for G and its complement over F (as in Definition 5.1.1), then the matrix (Ai,j · Bi,j) has nonze- ros on the diagonal and zero in every other entry, hence its rank is n. As it is a submatrix of the tensor product of A and B, its rank is at most the product of their ranks, proving the above inequality. On the other hand, [2] contains an example of a family of graphs Gn on n vertices satisfying o(1) mrFp (Gn), mrFq (Gn) ≤ n , where Fp and Fq are two distinct appropriately chosen prime fields (with p and q depending on n). An even more substantial gap between the min-rank of a graph over a finite field and its min-rank over the reals, at least when insisting on a representation by a positive semidefinite matrix, is given in [4], which provides an example of a sequence of graphs Gn on n vertices for which mrF(Gn) = 3 for some finite field F (depending on n), whereas the minimum possible rank over the reals by a positive semi-definite matrix is greater than n1/4.

• Haviv has recently combined the key lemma of [43] with the Lovász Local Lemma and proved a related result. To state it, we use the following density parameter. For a graph H with h ≥ 3 vertices, let m2(H) denote the maximum 0 f −1 ( 0 0) 0 value of h0−2 over all pairs h , f such that there is a subgraph H of H with h0 ≥ 3 vertices and f 0 edges. Theorem 5.5.2 (Haviv [49]). Let H be a graph with h ≥ 3 vertices and f edges. Then there is some c = c(H) > 0 so that for every ﬁnite ﬁeld F and every integer n there is a graph G on n vertices whose complement contains no copy of H, so that

n1−1/m2(H) mrF(G) ≥ c . log(n|F|)

Combining the proof of Haviv with our approach here we can get rid of the dependence on the size of the field and prove the following stronger result. Theorem 5.5.3. Let H be a graph with h ≥ 3 vertices. Then there is a constant c = c(H) > 0 such that for every finite or infinite field F and every integer n there is a graph G on n vertices whose complement contains no copy of H, so that

n1−1/m2(H) mrF(G) ≥ c . log n 5.5 concluding remarks 67

We omit the detailed proof.

• The results in this paper are formulated for undirected graphs, but can be easily extended to the directed case, with essentially the same proofs. In particular, Theorem 5.4.4 also holds for digraphs deﬁned by a polynomial (in this case we do not need to assume that the polynomial P is symmetric, see Subsection 4.4).

Acknowledgments. We thank Noga Alon, Lior Gishboliner, Adva Mond, and Frank Mousset for working with the author in order to obtain these results. We also thank Peter Nelson for telling us about his paper [72].

A APPENDIX

Here we give an improved bound for the parameter λ(n, C2t) (see Section 4.1.1) using the same approach as in Theorem 4.1.3. The argument is more complicated because Lemma 4.3.1 does not work for vertices at a distance of t from a given vertex. 1/(2t) Theorem A.0.1. For all t ≥ 2, we have λ(n, C2t) ≤ 12t n .

Proof. Let f be an orthonormal representation of G maximizing the largest eigenvalue of the corresponding Gram matrix M = M f . As in the proof of Theorem 4.1.3, 2t 2t it will sufﬁce to show that tr(M ) ≤ (12t) n. Recall the notations W(u0,..., uk) t and N (u0) introduced in the proof of Theorem 4.1.3. We have

2t tr M = ∑ W(u0,..., u2t−1, u0) , u0,u1,...,u2t−1∈V(G) where W(u0,..., u2t−1, u0) = 0 unless u0u1 ... u2t−1u0 forms a closed walk in G. Moreover, since G is C2t-free, any such closed walk must satisfy ui = uj for some t−1 0 ≤ i < j ≤ 2t − 1. Observe that this can happen either if u1,..., u2t−1 ∈ N (u0), or if d(u0, ui) = d(u0, u2t−i) = i for every i ∈ [t], and ui = u2t−i for some i ∈ [t − 1]. Thus if we deﬁne

Y(u0) = ∑ W(u0,..., u2t−1, u0) , t−1 u1,...,u2t−1∈N (u0)

Z(u0) = ∑ W(u0,..., u2t−1, u0) , u1,...,u2t−1: d(u0,ui ) = d(u0,u2t−i ) = i ∀i∈[t], ui = u2t−i for some i ∈ [t − 1]. then we have 2t tr M = ∑ (Y(u0) + Z(u0)) . u0∈V(G)

2t 2t We will show that Y(u0) ≤ (6t) and Z(u0) ≤ (4t) for every u0 ∈ V(G), which will complete the proof of the theorem. 2t To prove that Y(u0) ≤ (6t) for every vertex u0, one can repeat the argument from the proof of Theorem 4.1.3. We now turn to the task of upper-bounding Z(u0). For non-empty I ⊆ [t − 1], we deﬁne

ZI (u0) = ∑ W(u0,..., u2t−1, u0) , u1,...,u2t−1: d(u0,ui ) = d(u0,u2t−i ) = i ∀i∈[t], ui = u2t−i ∀i∈I.

69 70 appendix

and observe that by the inclusion-exclusion principle,

|I| Z(u0) = ∑ (−1) ZI (u0). I⊆[t−1], I6=∅

2t It thus sufﬁces to show that |ZI (u0)| ≤ (2t) for every non-empty I ⊆ [t − 1]. For vertices u0, u`, uk with d(u0, u`) = `, d(u0, uk) = k, where 0 ≤ ` < k ≤ t, deﬁne  2   S(u0, u`, uk) =  ∑ W(u`,..., uk) . u`+1,...,uk−1:  d(u0,ui ) = i.

Let I be a non-empty subset of [t − 1], and write I = {α1,..., αm−1}, where α1 < ... < αm−1. Also let α0 = 0 and αm = t. Now observe that

ZI (u0) m = ( ) ∑ ∏ S u0, uαi−1 , uαi uα1 ,...,uαm : i=1 ( )= d u0,uαi αi = ( ) ( ) ( ) ∑ S u0, u0, uα1 ∑ S u0, uα1 , uα2 ... ∑ S u0, uαm−1 , uαm . uα1 : uα2 : uαm : ( )= ( )= d(u0,uα )=αm d u0,uα1 α1 d u0,uα2 α2 m

Note that we have αi − αi−1 ≤ t − 1 for all i ∈ [m]. We shall show that 2(k−`) ∑ S(u0, u`, uk) ≤ (2t) uk : d(u0,uk )=k

for all k, ` and u0, u` such that d(u0, u`) = ` < k ≤ t and k − ` ≤ t − 1. Since it is clear by deﬁnition that S(u0, u`, uk) ≥ 0 for every u0, u`, uk, we may then conclude 2t that 0 ≤ ZI (u0) ≤ (2t) , as required. The remainder of the proof is very similar to the proof of Theorem 4.1.3. Given `, k and u0, u` as above, let Ai = {u ∈ V(G) : d(u0, u) = i, d(u`, u) = i − `} for ` < i ≤ k. Since d(u`, u) = i − ` ≤ t − 1 for all u ∈ Ai, we may apply Lemma 4.3.1 to conclude that χ(G[Ai]) ≤ 2t, and so we let {B(i, 1),..., B(i, 2t)} be a partition of Ai into 2t independent sets. Also observe that if d(u0, u`) = `, d(u0, ui) = i, and u0 ... u` ... ui is a walk in G, then d(u`, ui) = i − ` so that ui ∈ Ai. Therefore we obtain

∑ S(u0, u`, uk) uk : d(u0,uk )=k

= ∑ W(u`,..., uk, wk−1,..., w`+1, u`) u`+1,...,uk−1,uk : ui ∈Ai w`+1,...,wk−1: wi ∈Ai

= ∑ ∑ W(u`,..., uk, wk−1,..., w`+1, u`) . u ,...,u ,w ,...,w : a`+1,...,ak ∈[2t] `+1 k `+1 k−1 u ∈B(i,a ) b`+1,...,bk−1∈[2t] i i wi ∈B(i,bi ) appendix 71

Now observe that for all a`+1 ..., ak, b`+1,..., bk−1 ∈ [2t], we have

∑ W(u`,..., uk, wk−1,..., w`+1, u`) u`+1,...,uk,w`+1,...,wk−1: ui ∈B(i,ai ) wi ∈B(i,bi ) k `+1 ! | | | = ∑ f (u`) ∏ f (ui) f (ui) ∏ f (wi) f (wi) f (u`) u`+1,...,uk,w`+1,...,wk−1: i = `+1 i = k−1 ui ∈B(i,ai ) wi ∈B(i,bi )      k `+1 | | | = f (u`)  ∏  ∑ f (ui) f (ui)  ∏  ∑ f (wi) f (wi)  f (u`) i = `+1 ui ∈B(i,ai ) i = k−1 wi ∈B(i,bi )

= ( ) ( )| Thus if we deﬁne Pi,a ∑ui ∈B(i,a) f ui f ui , then we have

* k ! `+1 ! + S(u u u ) = f (u ) P P f (u ) ∑ 0, `, k ∑ ` , ∏ i,ai ∏ i,bi ` . uk : d(u0,uk )=k a`+1,...,ak ∈[2t] i=`+1 i=k−1 b`+1,...,bk−1∈[2t]

Note that Pi,a is an orthogonal projection onto the space spanned by { f (uj) : j ∈ B(i, a)}, and thus kPi,avk ≤ kvk for every vector v. It follows by Cauchy–Schwarz that

S(u , u , u ) ∑ 0 ` k uk : d(u0,uk )=k * ! ! + k `+1 ≤ f (u ), P P f (u ) ∑ ` ∏ i,ai ∏ i,bi ` a`+1,...,ak ∈[2t] i = `+1 i = k−1 b`+1,...,bk−1∈[2t] ! ! k `+1 ≤ k f (u )k · P P f (u ) ∑ ` ∏ i,ai ∏ i,ai ` a`+1,...,ak ∈[2t] i = `+1 i = k−1 b`+1,...,bk−1∈[2t] 2 2(k−`) ≤ ∑ k f (u`)k ≤ (2t) . a`+1,...,ak ∈[2t] b`+1,...,bk−1∈[2t]

2t As explained above, this completes the proof that 0 ≤ ZI (u0) ≤ (2t) for every vertex u0 and every non-empty I ⊆ [t − 1], which completes the proof of the theorem.

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