On $(t-1)$-colored paths in $t$-colored complete graphs

Abstract

Given $t$ distinct colors, we order the $t$ subsets of $t-1$ colors in some arbitrary manner. Let $G_1, G_2, \ldots , G_t$ be graphs. The $(t-1)$-chromatic Ramsey number, denoted by $r_{t-1}^t(G_1, G_2, \ldots, G_t)$, is defined to be the least number $n$ such that if the edges of the complete graph $K_n$ are colored in any fashion with $t$ colors, then for some $i$ the subgraph whose edges are colored with the $i$th subset of colors contains a $G_i$. In this paper, we find the value of $r_4^5(G_1, \ldots, G_5)$ when each $G_i$ is a path.