Set-coloring Ramsey numbers via codes

Abstract

For positive integers n, r, s with r > s, the set-coloring Ramsey number R(n; r, s) is the minimum N such that if every edge of the complete graph KN receives a set of s colors from a palette of r colors, then there is guaranteed to be a monochromatic clique on n vertices, that is, a subset of n vertices where all of the edges between them receive a common color. In particular, the case s = 1 corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on R(n; r, s) which imply that R(n; r, s) = 2 Θ(nr) if s/r is bounded away from 0 and 1. The upper bound extends an old result of Erdős and Szemerédi, who treated the case s = r − 1, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs.

Hypergraph Ramsey numbers of cliques versus stars

“…The dependence of the exponent of n on b, which comes from iterating Lemma 4.2, is inverse exponential. The authors suggested in problem 6.1 of [9] that stronger bounds than Lemma 4.2 might be true for the set-coloring Ramsey number R(n; r, s), especially in the regime s ≈ r − √ r where we are applying it here. Such improved upper bounds on R(n; r, s) would immediately improve the dependence on b in (5.1).…”

“…The lower bound is an obvious corollary of the lower bound in Theorem 1.3, while the upper bound involves some extra effort, in particular drawing on recent work of the authors on set-coloring Ramsey numbers [9]. One interesting corollary of this result is that there are positive constants c and c ′ such that 2 clog 2 n ≤ r(K (3) 5 − e, S (3)…”

“…The proof is essentially the result of iterating the argument for Theorem 1.3. However, we will also require a generalization of Lemma 4.1 from a recent paper of the authors [9]. Define the set-coloring Ramsey number R(n; r, s) to be the smallest positive integer N such that if every edge of K N receives a set of s colors from a palette of r colors, then there must exist a copy of K n where a single common color appears on every edge.…”

Tower Gaps in Multicolour Ramsey Numbers

“…We prove lower bounds on 𝑟 𝑘 (𝑡; 𝑞, 𝑝), thus giving lower bounds on certain set-colouring Ramsey numbers. However, we are not able to definitively resolve any questions from [4] due to central gaps in our understanding of hypergraph Ramsey numbers. See Section 5 for more on this.…”

“…A related notion called the set-colouring Ramsey number was introduced by Erdős, Hajnal and Rado in [9] and subsequently studied in [19] and much more recently in [4] and [2]. Borrowing notation from [4], let 𝑅 𝑘 (𝑡; 𝑞, 𝑠) denote the minimum number of vertices such that every (𝑞, 𝑠)-set colouring of 𝐾 (𝑘)…”