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

For two s -uniform hypergraphs H and F , the Turán number e x s (H , F) is the maximum number of edges in an F -free subgraph of H. Let s , r , k , n 1 , ... , n r be integers satisfying 2 ≤ s ≤ r and n 1 ≤ n 2 ≤ ⋯ ≤ n r . De Silva, Heysse and Young determined e x 2 (K n 1 , ... , n r , k K 2) and De Silva, Heysse, Kapilow, Schenfisch and Young determined e x 2 (K n 1 , ... , n r , k K r). 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 e x s (K n 1 , ... , n r (s) , k K s (s) ) for n 1 ≥ s 3 k 2 + s r. Then, using a probabilistic argument, we determine e x s (K n 1 , ... , n r (s) , k K r (s) ) for all n 1 ≥ k. Recently, Alon and Shikhelman determined asymptotically, for all F , the generalized Turán number e x 2 (K n , K s , F) , which is the maximum number of copies of K s in an F -free graph on n vertices. Here we determine e x 2 (K n 1 , ... , n r , K s , k K r) with n 1 ≥ k and n 3 = ⋯ = n r . Utilizing a result on rainbow matchings due to Glebov, Sudakov and Szabó, we determine e x 2 (K n 1 , ... , n r , K s , k K r) for all n 1 , ... , n r with n 4 ≥ r r (k − 1) k 2 r − 2 . [ABSTRACT FROM AUTHOR]