Extremal numbers of disjoint triangles in $r$-partite graphs

Abstract

For two graphs $G$ and $F$, the extremal number of $F$ in $G$, denoted by {ex}$(G,F)$, is the maximum number of edges in a spanning subgraph of $G$ not containing $F$ as a subgraph. Determining {ex}$(K_n,F)$ for a given graph $F$ is a classical extremal problem in graph theory. In 1962, Erdős determined {ex}$(K_n,k{K}_3)$, which generalized Mantel's Theorem. On the other hand, in 1974, {Bollobás}, Erdős, and Straus determined {ex}$(K_{n_1,n_2,\dots ,n_r},K_t)$, which extended Turán's Theorem to complete multipartite graphs. { In this paper,} we determine {ex}$(K_{n_1,n_2,\dots ,n_r},k{K}_3)$ for $r\ge 4$ and $10k-4\le n_1+4k\le n_2\le n_3\le \cdots \le n_r$.