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(Kn,F) for a given graph F is a classical extremal problem in graph theory. In 1962, Erdős determined ex(Kn,kK3), which generalized Mantel's Theorem. On the other hand, in 1974, Bollobás, Erdős, and Straus determined ex(Kn1,n2,…,nr,Kt), which extended Turán's Theorem to complete multipartite graphs. In this paper, we determine ex(Kn1,n2,…,nr,kK3) for r≥4 and 10k−4≤n1+4k≤n2≤n3≤⋯≤nr.