Tur\'{a}n Problems for Vertex-disjoint Cliques in Multi-partite Hypergraphs

For two $s$-uniform hypergraphs $H$ and $F$, the Tur\'{a}n number $ex_s(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Let $s, r, k, n_1, \ldots, n_r$ be integers satisfying $2\leq s\leq r$ and $n_1\leq n_2\leq \cdots\leq n_r$. De Silva, Heysse and Young determined $ex_2(K_{n_1, \ldots, n_r}, kK_2)$ and De Silva, Heysse, Kapilow, Schenfisch and Young determined $ex_2(K_{n_1, \ldots, n_r},kK_r)$. In this paper, as a generalization of these results, we consider three Tur\'{a}n-type problems for $k$ disjoint cliques in $r$-partite $s$-uniform hypergraphs. First, we consider a multi-partite version of the Erd\H{o}s matching conjecture and determine $ex_s(K_{n_1, \ldots, n_r}^{(s)},kK_s^{(s)})$ for $n_1\geq s^3k^2+sr$. Then, using a probabilistic argument, we determine $ex_s(K_{n_1, \ldots, n_r}^{(s)},kK_r^{(s)})$ for all $n_1\geq k$. Recently, Alon and Shikhelman determined asymptotically, for all $F$, the generalized Tur\'{a}n 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 $ex_2(K_{n_1, \ldots, n_r}, K_s, kK_r)$ with $n_1\geq k$ and $n_3=\cdots=n_r$. Utilizing a result on rainbow matchings due to Glebov, Sudakov and Szab\'{o}, we determine $ex_2(K_{n_1, \ldots, n_r}, K_s, kK_r)$ for all $n_1, \ldots, n_r$ with $n_4\geq r^r(k-1)k^{2r-2}$.