A lower bound for set-colouring Ramsey numbers

TL;DR

This paper establishes near-complete bounds for set-colouring Ramsey numbers, extending understanding beyond classical cases by introducing a new random colouring method to approximate these numbers for a wide range of parameters.

Contribution

The paper introduces a novel random colouring approach to determine set-colouring Ramsey numbers up to polylogarithmic factors for nearly all parameter ranges.

Findings

Determined $R_{r,s}(k)$ up to polylogarithmic factors for most $r$, $s$, and $k$.

Extended known bounds for set-colouring Ramsey numbers to broader parameter ranges.

Provided new insights into the asymptotic behavior of these Ramsey numbers.

Abstract

The set-colouring Ramsey number $R_{r,s}(k)$ is defined to be the minimum $n$ such that if each edge of the complete graph $K_n$ is assigned a set of $s$ colours from $\{1,\ldots,r\}$, then one of the colours contains a monochromatic clique of size $k$. The case $s = 1$ is the usual $r$-colour Ramsey number, and the case $s = r - 1$ was studied by Erd\H{o}s, Hajnal and Rado in 1965, and by Erd\H{o}s and Szemer\'edi in 1972. The first significant results for general $s$ were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstra\"ete, who showed that $R_{r,s}(k) = 2^{\Theta(kr)}$ if $s/r$ is bounded away from $0$ and $1$. In the range $s = r - o(r)$, however, their upper and lower bounds diverge significantly. In this note we introduce a new (random) colouring, and use it to determine $R_{r,s}(k)$ up to polylogarithmic factors in the exponent for essentially all $r$, $s$ and $k$.