Proof of a conjecture on induced subgraphs of Ramsey graphs

Abstract

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain "richness" properties characteristic of random graphs. More than 25 years ago, Erdős, Faudree and Sós conjectured that in any C-Ramsey graph there are Ω n 5/2 induced subgraphs, no pair of which have the same numbers of vertices and edges. Improving on earlier results of Alon, Balogh, Kostochka and Samotij, in this paper we prove this conjecture. Φ(G) = {e(H): H is an induced subgraph of G}.

Ramsey Numbers of Books and Quasirandomness

“…where in (14) we used that r ≤ k and (15) we used that 1/λ ≥ k2 4k and γ ≤ λ k 2 < λ k 2 −2 /2k, and in (16) we used our assumption that α r ≤ (k − r + 1)λ kr and that k 2 − r ≥ kr − k since r ≤ k. This is our desired bound.…”

“…Since our best lower bounds for r(K r ) come from random constructions, there have been many attempts to show that such C-Ramsey colorings exhibit properties that are typical for random colorings. For instance, Erdős and Szemerédi [7] proved that such colorings must have both red and blue densities bounded away from 0; Prömel and Rödl [17] proved that both the red and blue graphs contain induced copies of all "small" graphs (see also [8] for a simpler proof with better bounds); Jenssen, Keevash, Long, and Yepremyan [12] proved that both the red and blue graphs contain induced subgraphs exhibiting vertices with Ω(N 2/3 ) distinct degrees; and Kwan and Sudakov [14] proved that both the red and blue graphs contain Ω(N 5/2 ) induced subgraphs with distinct numbers of vertices and edges.…”

Anticoncentration for subgraph statistics

“…It is also interesting to study the probabilities Pr(X G,k = ℓ) for restricted classes of (hyper)graphs G. If these probabilities are small it would seem to give some evidence that the graphs in question are very "diverse" or "disordered". In particular, say that a graph is C-Ramsey if it has no clique or independent set of size C log 2 n. There has been a lot of work on diversity of Ramsey graphs from various points of view; in particular, Kwan and Sudakov [28] recently resolved a conjecture of Erdős, Faudree and Sós which effectively says that if G is an O(1)-Ramsey graph then for many values of k, the random variables X G,k have large support (see also [2,6,5,1,34,29] for related work). It would be very interesting to study the probabilities Pr(X G,k = ℓ) for Ramsey graphs.…”

“…In particular, say that a graph is ‐Ramsey if it has no clique or independent set of size . There has been a lot of work on diversity of Ramsey graphs from various points of view; in particular, Kwan and Sudakov recently resolved a conjecture of Erdős, Faudree, and Sós which effectively says that if is an ‐Ramsey graph, then for many values of , the random variables have large support (see also for related work). It would be very interesting to study the probabilities for Ramsey graphs.…”

Distinct degrees in induced subgraphs

“…Motivated by both the difficulty in providing explicit constructions and the challenge in improving the bounds for the Ramsey problem, an important theme of recent research in Ramsey theory has been establishing properties of Ramsey graphs supporting the intuition that they should be 'random-like'. This indirect study has been very fruitful, and it is now known that N -vertex Ramsey graphs display similar behaviour to the Erdős-Renyi random graph G N,1/2 in many respects: the edge density by Erdős and Szemerédi [9]; universality of small induced subgraphs by Prömel and Rödl [16]; the number of nonisomorphic induced subgraphs by Shelah [17]; the sizes and orders of induced subgraphs by Kwan and Sudakov [10,11] and Narayanan, Sahasrabudhe and Tomon [15].…”