Pseudorandom Ramsey Graphs Jacques Verstraete, UCSD [email protected] Abstract We outline background to a recent theorem [47] connecting optimal pseu- dorandom graphs to the classical off-diagonal Ramsey numbers r(s; t) and graph Ramsey numbers r(F; t). A selection of exercises is provided. Contents 1 Linear Algebra 1 1.1 (n; d; λ)-graphs . 2 1.2 Alon-Boppana Theorem . 2 1.3 Expander Mixing Lemma . 2 2 Extremal (n; d; λ)-graphs 3 2.1 Triangle-free (n; d; λ)-graphs . 3 2.2 Clique-free (n; d; λ)-graphs . 3 2.3 Quadrilateral-free graphs . 4 3 Pseudorandom Ramsey graphs 4 3.1 Independent sets . 4 3.2 Counting independent sets . 5 3.3 Main Theorem . 5 4 Constructing Ramsey graphs 5 4.1 Subdivisions and random blocks . 6 4.2 Explicit Off-Diagonal Ramsey graphs . 7 5 Multicolor Ramsey 7 5.1 Blowups . 8 1. Linear Algebra X. Since trace is independent of basis, for k ≥ 1: n If A is a square symmetric matrix, then the eigenvalues X tr(Ak) = tr(X−kAkXk) = tr(Λk) = λk: (2) of A are real. When A is an n by n matrix, we denote i i=1 them λ1 ≥ λ2 ≥ · · · ≥ λn. If the corresponding eigen- k vectors forming an orthonormal basis are e1; e2; : : : ; en, The combinatorial interpretation of tr(A ) as the num- n P then for any x 2 R , we may write x = xiei and ber of closed walks of length k in the graph G will be used frequently. When A is the adjacency matrix of a n n X 2 X graph G, we write λi(G) for the ith largest eigenvalue hAx; xi = λixi and hAx; yi = λixiyi: (1) i=1 i=1 of A and For any i 2 [n], note xi = hx; eii. Furthermore, A is di- λ(G) = maxfjλi(G)j : 2 ≤ i ≤ ng (3) −1 agonalizable: A = X ΛX where Λ has λ1; λ2; : : : ; λn on the diagonal and e1; e2; : : : ; en are the columns of and we refer to the eigenvalues of the graph rather than the eigenvalues of A. For X; Y ⊆ V (G), the number e(X; Y ) of ordered pairs (x; y) such that x 2 X, y 2 Y and fx; yg 2 E(G) is given by the inner product hAx; yi 1 1 Linear Algebra 2 in (1) where x and y are the characteristic vectors of Many more examples of (n; d; λ)-graphs are given in X and Y respectively { thus xv = 1 if v 2 X and the survey of Krivelevich and Sudakov [41]. For our xv = 0 otherwise. References on spectral graph theory purposes, we occasionally consider induced subgraphs include [15, 19, 33, 53]. of (n; d; λ)-graphs which are almost regular and whose eigenvalues are controlled by interlacing. 1.1. (n; d; λ)-graphs 1.2. Alon-Boppana Theorem If A is doubly stochastic, then e1 is the constant unit vector with eigenvalue equal to the common row sum. The infinite d-regular tree Td is the universal cover of In particular, if A is the adjacency matrix of a d-regular d-regular graphs, so for any d-regular graph G with n graph G, then λ1(G) = d and if λ(G) = λ, then the vertices, the number of closed walks of length 2k from graph is referred to as an (n; d; λ)-graph. The complete x to x is at least the number of closed walks of length graphs Kn are (n; n − 1; 1)-graphs and the complete 2k from the root of Td to the root of Td. This in turn 1 bipartite graphs Kn;n are (2n; n; n)-graphs. is at least 1 2k − 2 Moore graphs. A Moore graph of girth five is a d-regular d(d − 1)k−1: (7) k k − 1 graph with d2 +1 vertices and no cycles of length three or four. The diameter of such graphs is two, and if A The total number of walks of length 2k in G is at most is an adjacency matrix for such a graph, then d2k + (n − 1)λ2k by the trace formula, so A2 = J − A + (d − 1)I: (4) n2k − 2 d2k + (n − 1)λ2k ≥ d(d − 1)k−1: (8) k k − 1 Here J is the all 1 matrix and I is the identity matrix, 2 both with the same dimensions as A. Then λi + λi = Using this inequality, one can obtain a lower bound on d − 1 for all i ≥ 2 and so every eigenvalue other than λ. For fixed d, by selecting k to depend appropriately p 1 on n in (8), we obtain the Alon-Boppana Theorem [49]: the largest is 2 (1 ± 4d − 3). The Petersen graph is an example with d = 3 and the spectrum is 3115(−2)4. Theorem 1.1. Let d ≥ 1. If G is a d-regular n-vertex The non-existence of d-regular Moore graphs when d 62 n graph then f2; 3; 7; 57g was proved by Hoffman and Singleton [36] by considering integrality of the spectrum of the graph. p lim inf λ(Gn) ≥ 2 d − 1: (9) A brief survey of Moore graphs is found in [59]. n!1 A very short proof of the Alon-Boppana Theorem was Cayley graphs. Let Γ be a group and S ⊆ Γ closed found by Alon [49,50]. Ramanujan graphs are d-regular under inverse. A Cayley graph (Γ;S) has vertex set p graphs with λ ≤ 2 d − 1, and were constructed by Γ where fx; yg is an edge if xs = y for some s 2 S. Lubotzky, Philips and Sarnak [43] and Margulis [46] Let χ : γ 2 Γ denote the characters of Γ, then the γ as Cayley graphs, and more recently using polynomial eigenvalues of the Cayley graph interlacing by Marcus, Spielman and Srivastava [45]. X It turns out that random d-regular graphs [60] are also λγ = χγ (s): (5) s2S Ramanujan graphs with high probability, as proved in a major work by Friedman [27]. For example, if χ is the quadratic character of Fq, then the Paley graphs Pq have vertex set Fq where q = 1 1.3. Expander Mixing Lemma mod 4 and where fx; yg is an edge if χ(x − y) = 1 { this is a Cayley graph with generating set S equal to Let G be an (n; d; λ)-graph. If X; Y ⊂ V (G) are dis- joint and x and y are their characteristic vectors, then the set of non-zero quadratic residues of Fq, namely S = fx : χ(x) = 1g. Then the trivial character gives n 1 X λ1 = 2 (q − 1) as an eigenvalue, and the rest are given e(X; Y ) = hAx; yi = λixiyi: (10) by exponential sums. In particular i=1 1 This is the number of closed walks which return for the first X 2πijx λ(Pq) = max e q : (6) time to the root at time 2k. 1≤j≤q−1 x2S In particular, it was proved by Gauss [31] that λ(Pq) = 1 1 2 2 (q − 1). More generally, Weil's character sum in- equality [31] can be used to give an upper bound on λ(G) when G is a Cayley graph. 2 Extremal (n; d; λ)-graphs 3 If G is d-regular then λ1 = d and so by Cauchy- 2. Extremal (n; d; λ)-graphs Schwarz: 2.1. Triangle-free (n; d; λ)-graphs n X je(X; Y ) − dx1y1j ≤ λixiyi (11) Mantel's Theorem [44] states that a triangle-free graph i=2 with 2n vertices has at most n2 edges, with equal- 1 n 1 X 2 X 2 ity only for complete n by n bipartite graphs, which ≤ λ x2 y2 :(12) i i are triangle-free (2n; n; n)-graphs. In general, one may i=2 i=2 ask for triangle-free (n; d; λ)-graphs, where d; n and λ − 1 1 Since x1 = hx; e1i = n 2 jXj we get the expander mix- must satisfy (8), and in particular, λ = Ω(d 2 ). This 1 ing lemma of Alon [2]: leads to an extremal problem: when λ = cd 2 , what is the largest d for which there is a triangle-free (n; d; λ)- Theorem 1.2. Let G be an (n; d; λ)-graph and let X; Y ⊆ graph? If G is any triangle-free (n; d; λ)-graph with V (G). Then adjacency matrix A, then jXj 2e(X) − d jXj2 ≤ λjXj(1 − ) (13) 3 3 3 n n 0 = tr(A ) ≥ d − λ (n − 1) (18) 1 1 and if jXj = αn and jY j = βn then and so d ≤ λ(n − 1) 3 . If λ = O(d 2 ), then this gives 2 d = O(n 3 ). Alon [3] gave an ingenious construction of d p 2 e(X; Y ) − jXjjY j ≤ λ jXjjY j(1−α)(1−β): (14) 3 n a triangle-free Cayley (n; d; λ)-graph with d = Ω(n ) 1 and λ = O(n 3 ), showing the the above bounds are For a real α ≥ 0, a graph G with e(G) edges and tight. Other constructions were given by Kopparty [39] n density p = e(G)= 2 is α-pseudorandom if for every and Conlon [17], which are triangle-free (n; d; λ)-graphs 1 X ⊆ V (G), and shown to have λ = O(n 3 log n). jXj Theorem 2.1. There exist triangle-free (n; d; λ)-graphs e(X) − p ≤ αjXj: (15) 1 2 2 with λ = O(n 3 ) and d = Ω(n 3 ) as n ! 1.