Monochromatic Subgraphs in the Absence of a Properly Colored 4-Cycle

Abstract

Let k be a positive integer and let F and $H_{1}, H_{2}, \ldots , H_{k}$ be simple graphs. The proper-Ramsey number $pr_{k}(F; H_{1}, H_{2}, \ldots , H_{k})$ is the minimum integer n such that any k-coloring of the edges of $K_{n}$ contains either a properly colored copy of F or a copy of $H_{i}$ in color i, for some i. We consider the case where $F = C_{4}$ is fixed, and establish the exact value of the proper-Ramsey number when $\{H_i\}_{i=1}^k$ is a family containing only cliques, and nearly sharp bounds for the proper-Ramsey number when $\{H_i\}_{i=1}^k$ is a family containing only cycles or only stars. We also give a general bound for the proper-Ramsey number that is nearly tight when $\{H_i\}_{i=1}^k$ is a family of maximal split graphs.