Tur\'an number of disjoint triangles in 4-partite graphs

TL;DR

This paper determines the maximum edges in a 4-partite graph avoiding $k$ disjoint triangles when one part is large, and conjectures a similar bound for $r$-partite graphs avoiding disjoint cliques.

Contribution

It provides an exact extremal number for disjoint triangles in 4-partite graphs and proposes a conjecture for larger clique-free multipartite graphs.

Findings

Exact maximum edges for 4-partite graphs avoiding $k$ disjoint triangles

Conjecture on maximum edges for $r$-partite graphs avoiding $k$ disjoint cliques

Conditions on part sizes ensuring the bounds hold

Abstract

Let $k\ge 2$ and $n_1\ge n_2\ge n_3\ge n_4$ be integers such that $n_4$ is sufficiently larger than $k$. We determine the maximum number of edges of a 4-partite graph with parts of sizes $n_1,\dots, n_4$ that does not contain $k$ vertex-disjoint triangles. For any $r> t\ge 3$, we give a conjecture on the maximum number of edges of an $r$-partite graph that does not contain $k$ vertex-disjoint cliques $K_t$.