Complete subgraphs in a multipartite graph

Abstract

In 1975 Bollob\'as, Erd\H os, and Szemer\'edi asked the following question: given positive integers $n, t, r$ with $2\le t\le r-1$, what is the largest minimum degree $\delta(G)$ among all $r$-partite graphs $G$ with parts of size $n$ and which do not contain a copy of $K_{t+1}$? The $r=t+1$ case has attracted a lot of attention and was fully resolved by Haxell and Szab\'{o}, and Szab\'{o} and Tardos in 2006. In this paper we investigate the $r>t+1$ case of the problem, which has remained dormant for over forty years. We resolve the problem exactly in the case when $r \equiv -1 \pmod{t}$, and up to an additive constant for many other cases, including when $r \geq (3t-1)(t-1)$. Our approach utilizes a connection to the related problem of determining the maximum of the minimum degrees among the family of balanced $r$-partite $rn$-vertex graphs of chromatic number at most $t$.