Turán Problems for Vertex-disjoint Cliques in Multi-partite Hypergraphs

Abstract

Let s, r, k, n 1 , . . . , n r be integers satisfying 2 ≤ s ≤ r and n 1 ≤ n 2 ≤ • • • ≤ n r . For two s-uniform hypergraphs H and F , the Turán number ex s (H, F ) is the maximum number of edges in an F -free subgraph of H. De Silva, Heysse, and Young determined ex 2 (K n1,...,nr , kK 2 ) and De Silva, Heysse, Kapilow, Schenfisch and Young determined ex 2 (K n1,...,nr , kK r ). In this paper, we consider three Turán type problems for k disjoint cliques in r-partite s-graphs. First, we consider a multi-partite version of Erdős matching conjecture and determine ex s (K (s) n1,...,nr , kK (s) s ) for n 1 ≥ s 3 k 2 + sr. Using a probabilistic argument, we determine ex s (K (s) n1,...,nr , kK (s) r ) for all n 1 ≥ k. Recently, Alon and Shikhelman determined the generalized Turán number ex(K n , K s , F ) asymptotically for any F , which is the maximum number of copies of K s in an Ffree graph on n vertices. Utilizing Bollobás's Two Families Theorem, we determine ex(K n1,...,nr , K s , kK r ) when n 3 is sufficiently larger than k.