A Rainbow Erdös--Rothschild Problem

Abstract

We consider a multicolored version of a question posed by Erdos and Rothschild. For a fixed positive integer $r$ and a fixed graph $F$, we look for $n$-vertex graphs that admit the maximum number of $r$-edge colorings with the property that there is no copy of $F$ for which all edges are assigned different colors. We show that when $F$ is a bipartite graph with at least three edges and $r \geq 3$, the number of $r$-edge colorings of an extremal configuration is close to the number of such edge colorings of the complete graph $K_n$. On the other hand, for the rainbow pattern of $F=K_{k+1}$, the Turan graph $T_k(n)$ is the only extremal configuration for any $r\geq r_0(k)$ and large $n$.