No 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 Szemerédi 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 $k$.

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