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

For two s-uniform hypergraphs H and F, the Turan number ex(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 ex(2)(K-n1, ..., n(r), kK(2)) and De Silva, Heysse, Kapilow, Schenfisch and Young determined ex(2)(K-n1, ..., n(r), kK(r)). In this paper, as a generalization of these results, we consider three Turan-type problems for k disjoint cliques in r-partite s-uniform hypergraphs. First, we consider a multi-partite version of the Erd6s matching conjecture and determine ex(s)(K-n1((s)), ..., n(r), kK(s)((s))) for n(1) >= s(3)k(2) + sr. Then, using a probabilistic argument, we determine ex(s)(K-n1((s)), ..., n(r), kK(r)((s)) ) for all n(1) >= k. Recently, Alon and Shikhelman determined asymptotically, for all F, the generalized Turan number ex(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 ex(2)(K-n1, ..., n(r), K-s, kK(r)) with n(1) >= k and n(3) = ... = n(r). Utilizing a result on rainbow matchings due to Glebov, Sudakov and Szabo, we determine ex(2)(K-n1, ..., K-s, kK(r)) for all n(1), ..., n(r) with n(4) >= r(r) (k - 1)k(2r-2). (C) 2020 Elsevier B.V. All rights reserved.