Pseudorandom Ramsey Graphs

Jacques Verstraete, UCSD [email protected]

Abstract We outline background to a recent theorem [47] connecting optimal pseudorandom 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 ∈ 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 ∈ [n], note xi = hx, eii. Furthermore, A is di- λ(G) = max{|λi(G)| : 2 ≤ i ≤ n} (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 ∈ X, y ∈ Y and {x, y} ∈ 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 ∈ 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 √ 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 6∈ n graph then {2, 3, 7, 57} was proved by Hoffman and Singleton [36] by considering integrality of the spectrum of the graph. √ lim inf λ(Gn) ≥ 2 d − 1. (9) A brief survey of Moore graphs is found in [59]. n→∞ 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 √ graphs with λ ≤ 2 d − 1, and were constructed by Γ where {x, y} is an edge if xs = y for some s ∈ S. Lubotzky, Philips and Sarnak [43] and Margulis [46] Let χ : γ ∈ 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) s∈S 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 {x, y} 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 disjoint and x and y are their characteristic vectors, then the set of non-zero quadratic residues of Fq, namely S = {x : χ(x) = 1}. 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 x∈S

In particular, it was proved by Gauss [31] that λ(Pq) = 1 1 2 2 (q − 1). More generally, Weil’s character sum inequality [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 |e(X,Y ) − dx1y1| ≤ 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 |X| 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

|X| 2e(X) − d |X|2 ≤ λ|X|(1 − ) (13) 3 3 3 n n 0 = tr(A ) ≥ d − λ (n − 1) (18)

1 1 and if |X| = αn and |Y | = 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 ) − |X||Y | ≤ λ |X||Y |(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).

|X| Theorem 2.1. There exist triangle-free (n, d, λ)-graphs e(X) − p ≤ α|X|. (15) 1 2 2 with λ = O(n 3 ) and d = Ω(n 3 ) as n → ∞.

The maximum of the quantity on the left over all sub- Kopparty [39] gives a triangle-free (n, d, λ)-graph which 3 sets X is sometimes called the discrepancy of G. Note is a Cayley graph with vertex set F2t and generating that in the random graph Gn,p, the expected number set of edges in X is S = {(xy, xy2, xy3) : Trace(x) = 1} (19)

2 1 |X| where d = Ω(n 3 ) and λ = O(d 2 ), which is essentially (e(X)) = p (16) E 2 as good as the graphs of Alon [3]. and the Chernoff Bound can be used to show Gn,p Similar arguments show that if k is odd and G is a Ck- √ 1 1 is O( pn)-pseudorandom with high probability pro- free graph then d = O(λn k ) and when λ = O(d 2 ) this 2 vided p is not too small or not too large. A survey of gives d = O(n k ), and this too is tight [3, 41]. If F is a pseudorandom graphs is given by Krivelevich and Su- bipartite graph, then in many cases, the densest known dakov [41]. In particular, the expander mixing lemma n-vertex F -free graphs are graphs whose degrees are 1 shows that (n, d, λ)-graphs are λ-pseudorandom. Bol- all close to some number d and with λ = O(d 2 ) – see lobásand Scott [11] showed that if an n-vertex α- Fürediand Simonovits [30] for a survey of bipartite 2 pseudorandom graph has density p, where n ≤ p ≤ extremal problems. A particular rich source of such 3 2 p 2 1 − n , then α = Ω( p(1 − p)n ). The following con- graphs is the random polynomial model, due to Bukh jecture due to the author is open: and Conlon [13]. Conjecture A. Let G be an n-vertex graph of density p, 1 1 2.2. Clique-free (n, d, λ)-graphs where n ≤ p < 2 . Then for some X ⊆ V (G), If G is a K -free graph, then the common neighborhood 4 |X| 1 3 e(X) = p + Ω(p 2 n 2 ). (17) of any pair of adjacent vertices is an independent set. 2 If G is an (n, d, λ)-graph, this implies from (18) and (30) that We remark that with high probability, Gn,p satisfies this conjecture. When p = 1 and n = 2m is even, the 2 d3 − λ(n − 1) λn complete bipartite graph K has density p = 1 +o(1) ≤ α(G) ≤ . (20) m,m 2 e(G) d + λ 1 2 yet every set X of vertices has e(X) ≤ 4 |X| + O(n). 3 Pseudorandom Ramsey graphs 4

1 2 Consequently, d = O(λ 3 n 3 ). More generally, Sudakov, borhood of a vertex in Γ[q, s] induces Γ[q, s − 1], and q Szabo and Vu [55] showed Γ[q, 1] is empty with 2 vertices. Then an induction shows Γ[q, s] is Ks+1-free, and Γ[q, s] is an (n, d, λ)- 1 1 s−1 s−1 1− s−1 1− 1 1 d = O(λ n ). (21) graph with d = Θ(q 2 ) = Θ(n s ) and λ = Θ(d 2 ).

1 A geometric construction of K -free graphs H[k, s] when G is a Ks-free (n, d, λ)-graph. When λ = O(d 2 ) s+1 this gives was given by Alon and Krivelevich [4]: the vertex 1− 1 k d = O(n 2s−3 ). (22) set of H[k, s] is {1, 2, . . . , s} and vertices x and y are adjacent if their Hamming distance3 is larger than Constructing such optimal Ks-free graphs for s ≥ 4 is 2 k(1 − s(s+1) ). The n-vertex graphs H[k, s] are Ks+1- considered to be one of the major open problems in the free, but for every δ > 0 there exists k0 such that for area. k ≥ k0, every set X with Conjecture B. For s ≥ 4, there is a K -free (n, d, λ)- s 1− 2−δ 1 1− 1 s2(s+1)2 log s graph with λ = O(d 2 ) and d = Ω(n 2s−3 ). |X| > n (27)

Alon and Krivelevich [4] constructed Ks+1-free graphs induces a graph containing Ks. A careful analysis of as follows. Consider the one-dimensional subspaces the case s = 2 by Erd˝os[21] gives an explicit construc- s of Fq, and join two subspaces by an edge if they are tion of a lower bound for r(3, t), namely for each δ > 0, orthogonal to get a d-regular n-vertex (pseudo)graph 2 log 4 −δ G[q, s] with log 27 r(3, t) = Ω(t 8 ) (28) s qs − 1 n = = (23) where the exponent is roughly 1.1369. A more careful 1 q − 1 analysis of the construction of Alon and Krivelevich [4] s − 1 qs−1 − 1 d = = . (24) gives slightly better bounds than (27). 1 q − 1

Let S be the set of self-orthogonal 1-dimensional sub- 2.3. Quadrilateral-free graphs spaces. Then G[q, s] does not contain Ks+1 disjoint The “orthogonality” graphs above can also be used from S, since s+1 pairwise orthogonal one-dimensional to get extremal C4-free graphs: the graph G[q, 3] is subspaces outside S are not possible: if generators are quadrilateral-free and (q + 1)-regular with q2 + q + 1 x1, x2, . . . , xs+1, then c1x1 + ··· + csxs+1 = 0 with say vertices. This graph has q + 1 loops (corresponding to ci 6= 0, and then an inner product with xi gives the self-orthogonal subspaces), and removing the loops we 2 result. Furthermore, it is not hard to see using obtain a quadrilateral-free graph Gq with q +q+1 vertices and 1 q(q+1)2 edges. These are the orthogonal po- s − 1 s − 1 2 A2 = J + (d − )I (25) larity graphs of Erd˝os,Rényi and Sós[23]. Füredi[28] 1 1 proved that they are extremal C4-free graphs when 2 s−1 q > 13. Near extremal d-regular C6-free and C10-free that λ ≤ (d − 1 ) ≤ d. A computation shows 1 1− 1 graphs with λ = O(d 2 ) are constructed using general- G[q, s] − S has average degree d = Ω(n s−1 ) and 1 ized polygons – see [29, 42]. λ(G[q, s] − S) = O(d 2 ) by interlacing.

The above construction can be improved in a straight- 3. Pseudorandom Ramsey graphs forward way to give a Ks+1-free (n, d, λ) graph with 1− 1 1 d = Ω(n s ) and λ = O(d 2 ) – see Question 7. 3.1. Independent sets This matches a construction of Bishnoi, Ihringer and If G is an (n, d, λ)-graph with d ≥ 1 and X is an inde- Pepe [8], who define a graph Γ[q, s] using the bilinear Ps+1 pendent set in G, then from expander mixing form Q(x, y) = ξx1y1 + i=2 xiyi with ξ a quadratic non-residue in Fq. Let χ be the quadratic character of d 2 |X| |2e(X) − n |X| | ≤ λ|X|(1 − n ) (29) Fq. The vertex set of Γ[q, s] is 4 s+1 which gives the bound {hxi ∈ Fq : χ(Q(x, x)) = 1} (26) 1 nλ |X| ≤ . (30) d + λ and two one-dimensional subspaces hxi and hyi are ad- 3 jacent if χ(Q(x, y)) = 1. A key fact is that the neigh- The Hamming distance is d(x, y) = |{i : xi 6= yi}|. 4 In fact if µ is the least eigenvalue, one can similarly get the 2 r Hoffman-Delsarte Bound |X| ≤ −nµ/(d − µ) – see Godsil and These are the q-binomial coefficients. In general, k is the number of k-dimensional subspaces of an r-dimensional vector Newman [32] for this and other bounds. r r−1 r−k+1 k space over Fq, which is (q − 1)(q − 1) ... (q − 1)/(q − 1)(qk−1 − 1) ... (q − 1). 4 Constructing Ramsey graphs 5

n 2λn Let α(G) be the largest size of an independent set in a than d+1 . There are at most |Xr+1 ∩ Y | ≤ d graph G. We write α(X) for the largest independent choices of v ∈ Xr+1 ∩ Y . If v ∈ Xr+1\Y , then d set in a subset X of V (G). It is known that in G 1 the |Xr+1| ≤ (1 − )|Xr| by deﬁnition of Y , and so this n, 2 2n 1 largest eigenvalue is Θ(n) and λ = Θ(n 2 ). The above cannot happen for more than 1 bound α(Gn, 1 ) = O(n 2 ) whereas in fact α(Gn, 1 ) = 2 2 l2n log nm l t m O(log n). In fact G up until now gives the best lower s = = (36) n,p d log n bounds on Ramsey numbers r(s, t): using the Ks-free process [10] one gets for s ≥ 3: values of r. We conclude that the number of independent t-sets is at most s+1 t 2 1 r(s, t) = Ω (log t) s−2 (31) 1 t 2λnt−s log t ns . (37) t! s d whereas the best current upper bound on r(s, t) is A calculation using Stirling’s Formula gives the re- ts−1 quired bound. r(s, t) = O (32) (log t)s−2 We remark that we can get away with a similar count due to Shearer [51]. The eigenvalue bound (30) is of independent sets in n-vertex graphs G with average tight – see Question 11. It is a major conjecture that degree d and λ(G) ≤ λ.

α(Pq) = polylog(q) when q is a prime. Assuming the GRH, one can show α(Pq) = Ω((log q)(log log q)). 3.3. Main Theorem The following theorem [47] gives a new approach to 3.2. Counting independent sets Ramsey Theory via pseudorandom graphs: It turns out that counting independent sets is valuable Theorem 3.2. If there is an F -free (n, d, λ)-graph G for Ramsey Theory. By Turán’sTheorem [57], every 2 2 where λ ≥ log n and t = d 2n log n e, then graph of maximum degree d has an independent set of 4e2 d size at least t = n , and the number of such indepen- d+1 n log2n dent sets is at least r(F, t) > . (38) 4e2λ 1 n(n − d − 1)(n − 2d − 2) ··· = (d + 1)t. (33) Proof. We assume relevant quantities are integers for t! convenience. Select uniformly and randomly a subset 2 In (n, d, λ)-graphs, it turns out there are few indepen- log n X of G with each vertex having probability p = 4e2λ . dent sets that are much larger than this. The following The expected number of independent sets of size t in is a slight improvement [47] of a theorem of Alon and X is 2e2λ t Rödl[5]: pt = 2−t. (39) log2n Theorem 3.1. Let G be an (n, d, λ)-graph with d ≥ 1, 2 Since (|X|) = pn, there exists X ⊂ V (G) such that and let t ≥ 2n log n be an integer. Then the number of E d |X| ≥ pn − 2−t with α(X) < t. A computation gives independent sets of size t in G is at most the claimed bound on r(F, t). 2e2λ t We note that the proof actually tells us that with high 2 . (34) log n probability, a random subset of size about pn supplies Proof. Let X ⊂ V (G), and let the lower bound in (38). It would be very interesting to give explicit examples of this subset in an F -free d (n, d, λ)-graph. We remark that we can get away with a Y = {v ∈ V (G): |N(v) ∩ X| ≤ |X|}. (35) 2n similar result in n-vertex F -free graphs G with average degree d and λ(G) ≤ λ. By the expander mixing lemma, a calculation shows 4λ2n2 2λn |X||Y | ≤ 2 and therefore |X ∩ Y | ≤ . d d 4. Constructing Ramsey graphs Now suppose we have an independent set I = We now recall Conjecture B: suppose that we have a {v1, v2, . . . , vr} and sets X1,X2,...,Xr with X1 = V 1− 1 Ks-free (n, d, λ)-graph with d = Ω(n 2s−3 ) and λ = and r < t. Define Y as above with X = Xr and 1 1 2 O(d 2 ). Then for fixed s ≥ 3 and with t = 2n 2s−3 log n, let X = V \N(I) ⊆ X . Suppose we choose v to r+1 r Theorem 3.2 gives add to I, and note that v ∈ Xr+1. We aim to show that the number of choices of v is substantially less ts−1 r(s, t) = Ω (40) than the number of choices of vr if r is a bit larger log2s−4 t 4 Constructing Ramsey graphs 6

s+t−1 as t → ∞. We recall r(s, t) ≤ s−1 from the Erd˝os- and the above conjecture seem far beyond currently Szekeres Theorem [25], which for ﬁxed s and t → ∞ available techniques. gives ts−1 r(s, t) ≤ (1 + o(1)) . (41) 4.1. Subdivisions and random blocks (s − 1)! Let F be a graph and let P = (P1,P2,...,Pk) be a par- Shearer [51] improved this bound to tition of E(F ) into bipartite graphs with at least one ts−1 edge each. Let X = {x1, x2, . . . , xk} be new vertices, r(s, t) = O . (42) and let F be the graph with V (F ) = V (F ) ∪ X and logs−2 t P P edge set Thus the preceding lower bound (40) is tight up to log- k arithmic factors in t. Using the constructions discussed [ E(FP ) = {{xi, y} : y ∈ V (Pi)}. (46) in the section on clique-free pseudorandom graphs, one i=1 obtains via Theorem 3.2 for s ≥ 3: Let L(F ) be the family of all graphs FP taken over s t 2 r(s, t) = Ω (43) partitions P of E(F ) into paths with at least one edge (log t)s−2 each. For instance, when F is a triangle, then L(F )

consists of C4 plus a pendant edge and C6. If F is which is slightly worse than the bound (31) given by a pentagon then every member of L(F ) is a cycle of the random K -free process. s length at most ten plus a set of pendant edges. Let G

For F = Ck and k odd, optimal F -free (n, d, λ)-graphs be a bipartite graph with parts U and V containing no exist [3], and match or improve on the lower bound for member of L(F ) and such that every vertex of V has r(Ck, t) given by the Ck-free process. When F = Ck degree d. and k is even, it is a major open problem to deter- Theorem 4.1. Let F be a graph and let G be an L(F )- mine the order of magnitude of the extremal numbers free bipartite graph with parts U and V such that |U| = ex(n, F ) — the best lower bounds are due to Lazebnik, m and |V | = n and every vertex of V has degree d. If Ustimenko and Woldar [42]; a survey is given in [59]. dt > m + t log2 n, then When F = C4, the orthogonal polarity graphs Gq from Section 2.3 plus loops are (n, d, λ)-graphs with 2 1 r(F, t) > n. (47) n = q + q + 1, d = q + 1 and λ = q 2 . Mubayi and 3 Williford [48] checked that α(G ) = Θ(q 2 ) – in fact this q Proof. For each u ∈ U, let (Au,Bu) be a random par- matches the Hoffman-Delsarte Bound [36]. An upper tition of NG(u), independently for u ∈ U. Let H be 2 bound r(C4, t) < (t + 1) was given by Erd˝os,Faudree, the random graph with V (H) = V obtained by plac- Rousseau and Schelp [22], and improved to ing a complete bipartite graph with parts Au and Bu inside N (u) for each u ∈ U. It is evident that H is t2 G r(C , t) = O (44) F -free since G is L(F )-free. Now we show every inde- 4 (log t)2 pendent set in H has size less than t, which is sufficient by Szemerédi[56]. On the other hand, the orthogonal to show r(F, t) > n. Let I be a set of t vertices in H. 4 polarity graphs Gq provide r(C4, t) = Ω(t 3 ) whereas If |I ∩ NG(u)| = tu for u ∈ U, then 3 random graphs give r(C4, t) = Ω(t 2 ). The random 1−tu (e(I ∩ NG(u)) = 0) = 2 . (48) C4-free process analyzed by Bohman and Keevash [10] P 3 1 gives r(C4, t) = Ω(t 2 (log t) 2 ). If instead we use The- 1 Since the partitions (Au,Bu) are independent over dif- orem 3.2, then noting λ(G ) ∼ q 2 we get with t ∼ q ferent u ∈ U and the sum of t is at least dt, 8q(log q)2: u Y 2 (e(I) = 0) = (e(I ∩ NG(u)) = 0) q 3 1 P P 2 2 2 r(C4, t) = Ω 1 (log q) = Ω(t (log t) ) (45) u∈U q 2 Y = 21−tu which matches the C4-free process [10]. One of the u∈U notoriously difficult open conjectures of Erd˝os[16] is ≤ 2m−dt. as follows: There are n choices of I, so the expected number of 2− t Conjecture C. For some , T > 0, r(C4, t) ≤ t for independent sets of size t in H is all t ≥ T . n 2m−(q+1)t ≤ 2t log2 n+m−dt. (49) It is plausible that for some > 0 and all k ≥ 3, t 1− r(Ck, t) ≤ r(Ck−1, t) for large enough t, but this 5 Multicolor Ramsey 7

2 Since dt > m + t log2 n, this is less than 1, and so with regular bipartite graph H with n = q + q + 1 vertices positive probability, every independent set in H has in each part – we order the vertices in each part, say less than t. u1 < u2 < ··· < un in part U and v1 < v2 < ··· < vn in part V , and then form a graph H∗ as follows. We Random block constructions were used to give lower let V (H∗) = E(H) and let {u, v}, {w, x} ∈ E(H∗) if bounds on certain hypergraph Ramsey numbers [38]. u < w in U and v < x in V and {v, w} ∈ E(H). In Odd cycles. Let G be a bipartite graph of girth at least other words, we join the first and last edge of a path twelve with parts U and V of sizes m = (q+1)(q8 +q4 + of length three in H whose vertices in U and V are 1) and n = (q3 + 1)(q8 + q4 + 1) such that every vertex increasing. Kostochka, Pudlak and Rödl[40] refer to of V has degree q + 1 and every vertex of U has degree such a graph as a superline graph. The graph H∗ has 3 2 3 q + 1 – these are the incidence graphs of generalized (q + 1)(q + q + 1) = |E(H)| ∼ n 2 vertices. hexagons of order (q, q3) (see [34, 58] or [7] page 115 The independence number α(H∗) we claim is at most Corollary 5.38 for details about these constructions). 2n. To see this, note that an independent set of 2n Then G is L(F )-free and applying Theorem 4.1: vertices in H∗ corresponds to a set of 2n edges in H. 11 Amongst those 2n edges is a cycle, and that cycle con- r(C5, t) = Ω(t 8 ). (50) tains a path of length three in H whose vertices in U This is the first example of a graph F such that r(F, t) and V are increasing, contradicting the independence. has a higher exponent than that given by the random Therefore α(H∗) ≤ 2n. Furthermore, H∗ contains no ∗ F -free process. In a similar way, when F = C7, the triangles, since a triangle in H corresponds to three Ree-Tits octagons [34, 58] supply requisite bipartite edges {u1, v1}, {u2, v2} and {u3, v3} with u1 < u2 < u3 graphs G of girth at least sixteen with parts of sizes and v1 < v2 < v3 as well as the edges {v1, u2}, {v1, u3} m = (q + 1)(q9 + q6 + q3 + 1) and n = (q2 + 1)(q9 + and {v2, u3}. However, then {v1, u2, v2, u3} is the ver- q6 + q3 + 1) and all vertices in the larger part of degree tex set of a quadrilateral in H, a contradiction. So H∗ q + 1. Via Theorem 4.1: is triangle-free. While H∗ has roughly the same independence number as the triangle-free (n, d, λ)-graphs 11 r(C7, t) = Ω(t 9 ). (51) of Alon [3], it is not as good as those graphs in terms of spectral properties – see Question 20. Tetrahedron. If F = K4, then a C4-free bipartite graph G with parts of size n containing no 1-subdivision5 of Similar explicit constructions are given by Kostochka, Pudlak and Rödl[40], which show: K4 is L(F )-free. It is possible to show that |E(G)| = 7 O(n 5 ). If there is a d-regular n by n bipartite graph 8 5 2 r(4, t) = Ω(t 5 ) r(5, t) = Ω(t 3 ) r(6, t) = Ω(t ). (52) G containing no 1-subdivision of K4 with n vertices 2 and with d = Ω(n 5 ) even, then one can produce a ran- 4 It should be noted that these are far from the lower dom graph H where every vertex has degree Ω(n 5 ) and bounds supplied by random graphs, which have much ∗ 2 which may be an (n, d, λ)-graph with λ = O (n 5 ) (see larger exponents. We also note that the graph H∗ con- the case of triangles in Conlon [17]). Via Theorem 3.2, tains very large complete bipartite graphs, and seem 3−o(1) this would then show r(4, t) = t . However, the not to be amenable to an effective application of The- best construction of an n-vertex graph with no subdi- 4 orem 3.2. vision of K4 has only O(n 3 ) edges, coming from generalized quadrangles [58]. 5. Multicolor Ramsey 4.2. Explicit Off-Diagonal Ramsey graphs For a graph F and t ≥ 1, let rk(F, t) denote the Ramsey Alon [3] gave an explicit construction of a triangle- number r(F,F,...,F,t) with k colors (so k − 1 copies 2 3 free n-vertex graph with independence number O(n ). of F ). We consider r3(K3, t) = r(3, 3, t). Alon and This remains the best explicit construction for lower Rödl[5] solved an old conjecture of Erd˝osand Sós[24] 3 r(3,3,t) bounds on r(3, t), giving r(3, t) = Ω(t 2 ). A simple stating r(3,t) → ∞ as t → ∞, as follows: way to construct a triangle-free n-vertex graph with 2 ∗ 3 independence number O(n 3 ) was given by Pudlak, as Theorem 5.1. r(3, 3, t) = Θ (t ). follows. Proof. Take Alon’s triangle-free (n, d, λ)-graph Gn 2 1 0 Starting with the bipartite incidence graph of a projec- which has d = Θ(n 3 ) and λ = Θ(n 3 ), and let Gn tive plane of order q – this is a quadrilateral-free (q+1)- be a copy of Gn with the vertices randomly permuted. 5 We claim that with positive probability, A 1-subdivision S1(F ) of a graph F is the graph obtained by replacing each edge {u, v} of F with a path uwv. such that 1 the paths are required to be internally disjoint. Note S (F ) is 0 ∗ 3 1 α(Gn ∪ Gn) = O (t ). (53) bipartite and has |V (F )| + |E(F )| vertices. 5 Multicolor Ramsey 8

0 F To see this, pick a set X of t vertices of Gn ∪ Gn, V (G(r)) = v∈V (G) Xv and edges {x, y} : x ∈ Xu, y ∈ where t is as in Theorem 3.1, so that the number of Yv, {u, v} ∈ E(G). Kim and Mubayi [5] showed that independent sets of size t in Gn is at most blowing up a triangle-free graph from the triangle-free process does slightly better in the implicit logarithmic 2 t 4e λ factors since we can count the number of independent 2 . log n sets of a certain size in the blowup – see Question 15. In particular, one obtains (55) by taking a Ramsey graph The probability that a set like this is independent in 2 G for triangles versus K – this has Θ( t ) vertices – both G and G0 is t log t n n t and using G(r) with r = d log t e. 2 −1 4e λ 2t n The method of blowing up is eﬀective for F -free graphs 2 · . (54) log n t when F is not bipartite, but does not work when F is bipartite (since indeed the blowup contains large com- Using standard estimates on binomial coeﬃcients, if n plete bipartite graphs). is large enough, this is less than 1, and therefore with 0 positive probability Gn ∪ Gn has no independent set of ∗ 1 Exercises size t = O (n 3 ).

More precisely, the proof above shows 1. Let k be a positive even integer. Prove that if A is a symmetric pointwise non-negative matrix, and x is 3 t a pointwise non-negative unit vector, then Akx, x ≥ r(3, 3, t) = Ω 4 . (55) (log t) hAx, xik.6

The proof is not too far from best possible, since it is 1 1 2. Prove that λ(P ) = (q 2 − 1). possible to show that if G is a graph that is a union of q 2 two n-vertex triangle-free graphs with average degree 2 3. For d-regular Moore graphs of girth 5, show d ∈ O(n 3 ), then by adapting the proofs of Shearer [51]: {2, 3, 7, 57}. 1 3 α(G) = Ω(n log n) (56) 1 1 4. Let n ≥ 1 and let n ≤ p < 2 . Prove that with high and consequently we could not get a better lower bound probability, every X ⊆ V (Gn,p) has t 3 than r(3, 3, t) = Ω(( log t ) ) using the above approach. |X| 1 3 The general upper bound for r(3, 3, t) is based on re- e(X) − p = Θ(p 2 n 2 ). 2 sults of Shearer [51], as follows:

t3 5. Suppose there is a triangle-free (n, d, λ)-graph and r(3, 3, t) = O . (57) (log t)2 let k be a positive integer. Prove that there is a triangle-free (kn, kd, kλ)-graph. The following conjecture is plausible: 1 Conjecture D. 6. Derive (9) from (8), and show that for d ≤ 2 n, 1 λ = Ω(d 2 ) as d → ∞. t3 r(3, 3, t) = Θ 2 . (58) (log t) 7. Let s ≥ 2. Prove that there is a Ks+1-free (n, d, λ)- 1− 1 1 graph with d = Ω(n s ) and λ = O(d 2 ) using k- ks+k−1 Using known constructions of (n, d, λ)-graphs, Alon dimensional spaces in Fq . and R¨odl[5] gave lower bounds on rk(F, t) for a va- riety of bipartite graphs F . Quite remarkably, it turns 8. Let q be a suitable7 prime power and out that for all k ≥ 4, 2 3 q 2q 3 S = {(xy, xy , xy ): 3

loops. Then ﬁnd e(Gq) and λ(Gq). 5.1. Blowups 6 It is more challenging to prove this “Matrix H¨older”inequality when k is odd. For r ≥ 1, a graph G, and disjoint sets Xv : v ∈ V (G), 7 For which prime powers does the proof work? the r-blowup of G is a graph G(r) with vertex set 5 Multicolor Ramsey 9

10. Prove that (n, d, λ)-graphs are λ-pseudorandom. References

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