Ramsey Graphs Induce Subgraphs of Quadratically Many Sizes

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. Motivated by an old problem of Erdős and McKay, recently Narayanan, Sahasrabudhe and Tomon conjectured that for any fixed C, every n-vertex C-Ramsey graph induces subgraphs of Θ n 2 different sizes. In this paper we prove this conjecture.

Proof of a conjecture on induced subgraphs of Ramsey graphs

“…In fact, recent developments bounding |Φ(G)| due to Narayanan, Sahasrabudhe and Tomon [30], and ourselves [27], make this idea seem even more promising. In [30], the authors made the simple observation (using the pigeonhole principle) that in any n-vertex graph G, there is a set A of √ n vertices with degrees lying in an interval of length √ n. If G is diverse, and U is a random vertex set of linear size, then the degrees d U (x), for x ∈ A, are likely to take n 1/2−o(1) different values, very tightly packed in an interval of length O( √ n).…”

“…By augmenting U with different combinations of vertices in A, we can obtain subgraphs of many different sizes, all lying in a fixed interval of length O(n). Adapting these ideas to our context, and using the further refinements in [27], one can prove that we can actually obtain Ω(n) values of e(G[U ∪ Y ]) among subsets Y ⊆ A of a certain fixed size, tightly packed in an interval of length O(n). So, as a rough plan to prove Theorem 1.1, one might start with vertex subsets W − , W + of fixed size ℓ = Θ(n) such that e(W + ) − e(W − ) = Ω n 3/2 , provided by a discrepancy theorem.…”

“…In [10] Bukh and Sudakov introduced the notion of diversity: an n-vertex graph is said to be diverse if |N (x)△N (y)| = Ω(n) for most pairs of distinct vertices x, y. We will need a slightly stronger notion than diversity, which we introduced in [27]. Say an n-vertex graph is (δ, ε)-rich if for any vertex…”

“…The next lemma appears as [27,Lemma 4], showing that Ramsey graphs contain large rich induced subgraphs. Lemma 3.1.…”

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.…”