An extension of the rainbow Erd\H{o}s-Rothschild problem

TL;DR

This paper extends the rainbow Erd ext{-}Rothschild problem by characterizing the extremal graphs that maximize the number of certain edge-colorings avoiding large rainbow cliques, identifying Turán graphs as optimal.

Contribution

It proves that for large graphs and sufficient colors, Turán graphs uniquely maximize the count of specific rainbow-free colorings, extending previous extremal results.

Findings

Turán graphs are optimal for large n and r in avoiding rainbow K_k with s or more colors.

The Turán graph T_{k-1}(n) uniquely maximizes the number of free r-colorings.

The result generalizes the rainbow Erd
51s-Rothschild problem to broader parameters.

Abstract

Given integers $r \geq 2$, $k \geq 3$ and $2 \leq s \leq \binom{k}{2}$, and a graph $G$, we consider $r$-edge-colorings of $G$ with no copy of a complete graph $K_k$ on $k$ vertices where $s$ or more colors appear, which are called $\mathcal{P}_{k,s}$-free $r$-colorings. We show that, for large $n$ and $r \geq r_0(k,s)$, the $(k-1)$-partite Tur\'an graph $T_{k-1}(n)$ on $n$ vertices yields the largest number of $\mathcal{P}_{k,s}$-free $r$-colorings among all $n$-vertex graphs, and that it is the unique graph with this property.