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- W2968832023 abstract "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 $s, r, k, n_1, ldots, n_r$ be integers satisfying $2leq sleq r$ and $n_1leq n_2leq cdotsleq n_r$. 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 Turan-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 $ex_s(K_{n_1, ldots, n_r}^{(s)},kK_s^{(s)})$ for $n_1geq s^3k^2+sr$. Then, using a probabilistic argument, we determine $ex_s(K_{n_1, ldots, n_r}^{(s)},kK_r^{(s)})$ for all $n_1geq k$. 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 $ex_2(K_{n_1, ldots, n_r}, K_s, kK_r)$ with $n_1geq k$ and $n_3=cdots=n_r$. Utilizing a result on rainbow matchings due to Glebov, Sudakov and Szabo, we determine $ex_2(K_{n_1, ldots, n_r}, K_s, kK_r)$ for all $n_1, ldots, n_r$ with $n_4geq r^r(k-1)k^{2r-2}$." @default.
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- W2968832023 date "2019-08-16" @default.
- W2968832023 modified "2023-09-27" @default.
- W2968832023 title "Tur'{a}n Problems for Vertex-disjoint Cliques in Multi-partite Hypergraphs" @default.
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