Tur\'an number of complete multipartite graphs in multipartite graphs

TL;DR

This paper extends the Erdős–Stone theorem to multipartite graphs, determining exact extremal edge counts for certain complete multipartite subgraphs and characterizing extremal structures.

Contribution

It provides exact values of the extremal function for specific parameters and characterizes extremal graphs when divisibility conditions are met.

Findings

Exact extremal edge counts for t ≤ 3, r < k ≤ 2r, large n

Characterization of extremal graphs when r divides k

Extension of Erdős–Stone theorem to multipartite graphs

Abstract

In this paper we study a multi-partite version of the Erd\H{o}s--Stone theorem. Given integers $r<k$ and $t\ge 1$, let $\text{ex}_k(n, K_{r+1}(t))$ be the maximum number of edges of $K_{r+1}(t)$-free $k$-partite graphs with $n$ vertices in each part, where $K_{r+1}(t)$ is the complete $(r+1)$-partite graph with $t$ vertices in each part. We determine the exact value of $\text{ex}_k(n, K_{r+1}(t))$ for $t\le 3$, $r<k\le 2r$ and sufficiently large $n$. We also characterize all extremal graphs for $r, k$ such that $r$ divides $k$, analogous to a result of Erd\H os and Simonovits on forbidding $K_{r+1}(t)$ in general graphs.