We present a new proof of the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem that $ex(n, K_{s,t}) = O(n^{2-1/t})$ for $s, t \geq 2$. The new proof is elementary, avoiding the use of convexity. For any $d$-uniform hypergraph $H$, let $ex_d(n,H)$ be the maximum possible number of edges in an $H$-free $d$-uniform hypergraph on $n$ vertices. Let $K_{H, t}$ be the $(d+1)$-uniform hypergraph obtained from $H$ by adding $t$ new vertices $v_1, \dots, v_t$ and replacing every edge $e$ in $E(H)$ with $t$ edges $e \cup \left\{v_1\right\},\dots, e \cup \left\{v_t\right\}$ in $E(K_{H, t})$. If $H$ is the $1$-uniform hypergraph on $s$ vertices with $s$ edges, then $K_{H, t} = K_{s, t}$. We prove that $ex_{d+1}(n,K_{H,t}) = O(ex_d(n, H)^{1/t} n^{d+1-d/t} + t n^d)$ for any $d$-uniform hypergraph $H$ with at least two edges such that $ex_d(n, H) = o(n^d)$. Thus $ex_{d+1}(n,K_{H,t}) = O(n^{d+1-1/t})$ for any $d$-uniform hypergraph $H$ with at least two edges such that $ex_d(n, H) = O(n^{d-1})$, which implies the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem in the $d = 1$ case. This also implies that $ex_{d+1}(n, K_{H,t}) = O(n^{d+1-1/t})$ when $H$ is a $d$-uniform hypergraph with at least two edges in which all edges are pairwise disjoint, which generalizes an upper bound proved by Mubayi and Verstra\"{e}te (JCTA, 2004). We also obtain analogous bounds for 0-1 matrix Tur\'{a}n problems.

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K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n 定理的推广

我们提出了 K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n 定理的新证明，即 $ex(n, K_{s,t}) = O(n^{2-1/t})$ 对于 $s, t \geq 2$。新的证明是基本的，避免使用凸性。对于任何 $d$-uniform hypergraph $H$，让 $ex_d(n,H)$ 是 $n$ 顶点上的 $h$-free $d$-uniform hypergraph 中的最大可能边数。令 $K_{H, t}$ 是通过添加 $t$ 新顶点 $v_1, \dots, v_t$ 并替换 $ 中的每条边 $e$ 从 $H$ 获得的 $(d+1)$-uniform hypergraph E(H)$ 与 $t$ 边 $e \cup \left\{v_1\right\},\dots, e \cup \left\{v_t\right\}$ in $E(K_{H, t} )$。如果$H$是$s$顶点上$s$边的$1$-uniform超图，则$K_{H, t} = K_{s, t}$。我们证明 $ex_{d+1}(n,K_{H,t}) = O(ex_d(n, H)^{1/t} n^{d+1-d/t} + tn^d )$ 用于任何具有至少两条边的 $d$-统一超图 $H$，使得 $ex_d(n, H) = o(n^d)$。因此 $ex_{d+1}(n,K_{H,t}) = O(n^{d+1-1/t})$ 对于任何具有至少两条边的 $d$-均匀超图 $H$使得 $ex_d(n, H) = O(n^{d-1})$，这意味着 K\H{o}v\'{a}ri-S\'{o}s-Tur\' {a}n 定理在 $d = 1$ 的情况下。这也意味着 $ex_{d+1}(n, K_{H,t}) = O(n^{d+1-1/t})$ 当 $H$ 是 $d$-uniform hypergraph 时至少有两条边，其中所有边都是成对不相交的，这概括了 Mubayi 和 Verstra\"{e}te (JCTA, 2004) 证明的上限。我们还获得了 0-1 矩阵 Tur\'{a} 的类似边界n 问题。