The Size Multipartite Ramsey Numbers for Paths

Abstract

For graphs $G_1,G_2,\dots ,G_k$, the (generalized) \text{size multipartite Ramsey number} $m_j(G_1,G_2,\dots ,G_k)$ is the least natural number $m$ such that any coloring of the edges of $K_{j\times m}$ with $k$ colors will yield a copy of $G_i$ in the $i$th color for some $i$. In this note, we determine the exact value of the size multipartite Ramsey number $m_j(P_s,P_t)$ for $s=2,3$ and all integers $t\ge 2$, where $P_t$ denotes a path on $t$ vertices.