Long monochromatic paths and cycles in 2-edge-colored multipartite graphs

Abstract

We solve four similar problems: for every fixed $s$ and large $n$, we describe all values of $n_1,\dots ,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\dots ,n_s}$ there exists a monochromatic (i) cycle $C_{2n}$ with $2n$ vertices, (ii) cycle $C_{\ge 2n}$ with at least $2n$ vertices, (iii) path $P_{2n}$ with $2n$ vertices, and (iv) path $P_{2n+1}$ with $2n+1$ vertices.

This implies a generalization for large $n$ of the conjecture by Gyárfás, Ruszinkó, Sárközy and Szemerédi that for every $2$-edge-coloring of the complete $3$-partite graph $K_{n,n,n}$ there is a monochromatic path $P_{2n+1}$. An important tool is our recent stability theorem on monochromatic connected matchings.