A generalization of the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem

Abstract

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.