THE NUMBER OF EDGE COLORINGS WITH NO MONOCHROMATIC CLIQUES

Abstract

Let $F(n,r,k)$ denote the maximum possible number of distinct edge-colorings of a simple graph on $n$ vertices with $r$ colors which contain no monochromatic copy of $K_k$. It is shown that for every fixed $k$ and all $n\,{>}\,n_0(k)$, $F(n,2,k)\,{=}\,2^{t_{k-1}(n)}$ and $F(n,3,k)\,{=}\,3^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the maximum possible number of edges of a graph on $n$ vertices with no $K_k$ (determined by Turán's theorem). The case $r\,{=}\,2$ settles an old conjecture of Erdős and Rothschild, which was also independently raised later by Yuster. On the other hand, for every fixed $r\,{>}\,3$ and $k\,{>}\,2$, the function $F(n,r,k)$ is exponentially bigger than $r^{t_{k-1}(n)}.$ The proofs are based on Szemerédi's regularity lemma together with some additional tools in extremal graph theory, and provide one of the rare examples of a precise result proved by applying this lemma.