Complete subgraphs in a multipartite graph

TL;DR

This paper investigates the maximum minimum degree in r-partite graphs avoiding complete subgraphs, resolving the problem exactly for certain cases and approximately for others, extending over forty years of open questions.

Contribution

It provides the first exact solutions for the problem when r ≡ -1 mod t and approximate bounds for many other cases, including large r relative to t.

Findings

Exact solution for r ≡ -1 mod t case.

Approximate bounds for r ≥ (3t-1)(t-1).

Connection to maximum minimum degree in balanced r-partite graphs.

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 $t$.