A lower bound for set-coloring Ramsey numbers.

Abstract

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