Ramsey Set Numbers in Balanced Complete Multipartite Graphs.

Abstract

One natural extension of classical Ramsey numbers to multipartite graphs is to consider 2-colorings of the complete multipartite graph consisting of $n$ parts, each of size $k$, denoted $K_{n\times k}$. We may then ask for the minimum integer $n$ such that $K_{n\times k}\to (G,H)$ for two given graphs $G$ and $H$. We study this number for the cases when $G$ and $H$ are paths or cycles and show some general bounds and relations to classical Ramsey theory.