Turán number of complete multipartite graphs in multipartite graphs
Jie Han, Yi Zhao
TL;DR
This work advances the extremal theory for multipartite host graphs by determining the exact Turán-type quantity ex_k(n, K_{r+1}(t)) for t in {2,3} when r<k≤2r and n large. Building on the Erdős–Stone paradigm, the authors introduce multipartite Zarankiewicz numbers and a lower-bound construction g(n,r,k,t), then prove matching upper bounds via a robust stability framework that ties extremal graphs to Turán-type templates. When r divides k, they provide a near-complete structural characterization of extremal graphs, including an explicit partition with at most a constant number of exceptional vertices, paralleling Erdős–Simonovits results in the general setting. The results illuminate how multipartite restrictions influence extremal configurations and raise natural conjectures for larger t and k, linking Turán-type behavior to Zarankiewicz-type bounds in a multipartite context.
Abstract
In this paper we study a multi-partite version of the Erdős--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.