Constrained Ramsey numbers for rainbow $P_5$

Abstract

For any graph $H$, let $R_k(H)$ denote the minimum integer $n$ such that in every $k$-edge-coloring of $K_n$, there is a monochromatic copy of $H$. Given two graphs $H$ and $G$, the \emph{constrained Ramsey number} $f(H,G)$ is the minimum integer $n$ such that in every edge-coloring of $K_n$ with any number of colors, there is either a monochromatic copy of $H$ or a rainbow copy of $G$. Let $P_t$ be the path on $t$ vertices. Gyárfás, Lehel and Schelp proved that $f(H,P_5)=R_3(H)$ when $H$ is a path, a cycle, or a connected non-bipartite graph. Li, Besse, Magnant, Wang and Watts conjectured that $f(H,P_5)=R_3(H)$ for any graph $H$, and confirmed this for every connected graph and every bipartite graph. In this paper, we address this conjecture for several classes of disconnected graphs with chromatic number at least 3. To the best of our knowledge, our general results encompass all previously known results of this type. We also obtain several results for a bipartite variant of the problem. In addition, we propose a series of related questions from several directions for further research. Our proofs combine the structural characterization of Thomason and Wagner with Simonovits' decomposition-family method and Ramsey numbers for families of graphs.

arXiv comment29 pages