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arXiv · 2405.16561

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

Jie Han·Yi Zhao
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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.

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Jie Han, Yi Zhao. 2024-05-26. Tur\'an number of complete multipartite graphs in multipartite graphs. https://arxiv.org/abs/2405.16561

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