Turán problems for vertex-disjoint cliques in multi-partite hypergraphs

Abstract

For two s-uniform hypergraphs H and F, the Turán number exs(H,F) is the maximum number of edges in an F-free subgraph of H. Let s,r,k,n1,…,nr be integers satisfying 2≤s≤r and n1≤n2≤⋯≤nr. De Silva, Heysse and Young determined ex2(Kn1,…,nr,kK2) and De Silva, Heysse, Kapilow, Schenfisch and Young determined ex2(Kn1,…,nr,kKr). In this paper, as a generalization of these results, we consider three Turán-type problems for k disjoint cliques in r-partite s-uniform hypergraphs. First, we consider a multi-partite version of the Erdős matching conjecture and determine exs(Kn1,…,nr(s),kKs(s)) for n1≥s3k2+sr. Then, using a probabilistic argument, we determine exs(Kn1,…,nr(s),kKr(s)) for all n1≥k. Recently, Alon and Shikhelman determined asymptotically, for all F, the generalized Turán number ex2(Kn,Ks,F), which is the maximum number of copies of Ks in an F-free graph on n vertices. Here we determine ex2(Kn1,…,nr,Ks,kKr) with n1≥k and n3=⋯=nr. Utilizing a result on rainbow matchings due to Glebov, Sudakov and Szabó, we determine ex2(Kn1,…,nr,Ks,kKr) for all n1,…,nr with n4≥rr(k−1)k2r−2.