A Density Tur\'an Theorem

TL;DR

This paper extends a classical density Turán theorem to multipartite graphs, characterizing the minimum edge density needed to guarantee a subgraph, with specific structural conditions on the subgraph.

Contribution

It establishes a precise density threshold for multipartite graphs to contain a given subgraph, generalizing and extending previous results for cliques.

Findings

Determines the exact density threshold for multipartite graphs to contain a subgraph H.

Characterizes the structure of H for the threshold to hold, involving vertex coloring and matchings.

Recovers known results for cliques and extends to broader classes of graphs.

Abstract

Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. For such a graph $F,$ a classical result of Simonovits from 1966 shows that every graph on $n\ge n_0(F)$ vertices with more than $\frac{\chi(F)-2}{\chi(F)-1}\cdot \frac{n^2}{2}$ edges contains a copy of $F$. In this paper we derive a similar theorem for multipartite graphs. For a graph $H$ and an integer $\ell \geq v(H)$, let $d_{\ell}(H)$ be the minimum real number such that every $\ell$-partite graph whose edge density between any two parts is greater than $d_{\ell}(H)$ contains a copy of $H$. Our main contribution is to show that $d_{\ell}(H)=\frac{\chi(H)-2}{\chi(H)-1}$ for $\ell \ge \ell_0(H)$ sufficiently large if and only if $H$ admits a vertex-colouring with $\chi(H)-1$ colours such that all colour classes but one are independent sets, and the exceptional class induces just a matching. When $H$ is a clique, this recovers a result of Pfender [Complete subgraphs in multipartite graphs, Combinatorica 32 (2012), 483--495]. We also consider several extensions of Pfender's result.