Complete subgraphs in multipartite graphs

TL;DR

This paper establishes a Turan-type theorem for multipartite graphs, determining the minimal edge density needed to guarantee complete subgraphs, and provides structural insights into extremal graphs, including a counterexample to a previous conjecture.

Contribution

It extends Turan's theorem to l-partite graphs, finds the minimal edge density for large l, and disproves a conjecture regarding triangle densities.

Findings

For large l, the minimal density is (k-2)/(k-1).

The structure of extremal graphs is characterized.

Disproves the conjecture that d^3_{13} equals 1/3, showing it is 1/2.

Abstract

Turan's Theorem states that every graph of a certain edge density contains a complete graph $K^k$ and describes the unique extremal graphs. We give a similar Theorem for l-partite graphs. For large l, we find the minimal edge density $d^k_l$, such that every $\ell$-partite graph whose parts have pairwise edge density greater than $d^k_l$ contains a $K^k$. It turns out that $d^k_l=(k-2)/(k-1)$ for large enough l. We also describe the structure of the extremal graphs. For the case of triangles we show that $d^3_{13}=1/2$, disproving a conjecture by Bondy, Shen, Thomasse and Thomassen.