Tight bounds towards Zarankiewicz problem in hypergraph
Guorong Gao, Jianfeng Hou, Shuping Huang, Hezhi Wang
TL;DR
The paper extends the Zarankiewicz problem to $r$-partite $r$-graphs by studying $z(m_1,\ldots,m_r;s_1,\ldots,s_r)$, the maximum edge count avoiding an ordered $K_{s_1,\ldots,s_r}$. It develops tight, parameter-sensitive bounds using a hypergraph version of the random algebraic method, establishing a lower bound in a broad regime and an inductive upper bound, thereby proving $z(m_1,\ldots,m_{r-1},n;s_1,\ldots,s_{r-1},t)=\Theta\left(m_1\cdots m_{r-1} n^{1-1/(s_1\cdots s_{r-1})}\right)$ under suitable conditions. The work generalizes Conlon's 2-graph results to hypergraphs and demonstrates that the random algebraic approach yields near-tight Zarankiewicz bounds for diverse hypergraph configurations. Its findings advance extremal hypergraph theory by providing explicit asymptotics in a nontrivial multi-parameter setting with potential applications to related combinatorial constructions.
Abstract
The classical Zarankiewicz problem, which concerns the maximum number of edges in a bipartite graph without a forbidden complete bipartite subgraph, motivates a direct analogue for hypergraphs. Let $K_{s_1,\ldots, s_r}$ be the complete $r$-partite $r$-graph such that the $i$-th part has $s_i$ vertices. We say an $r$-partite $r$-graph $H=H(V_1,\ldots,V_r)$ contains an ordered $K_{s_1,\ldots, s_r}$ if $K_{s_1,\ldots, s_r}$ is a subgraph of $H$ and the set of size $s_i$ vertices is embedded in $V_i$. The Zarankiewicz number for $r$-graph, denoted by $z(m_1, \ldots, m_{r}; s_1,, \ldots,s_{r})$, is the maximum number of edges of the $r$-partite $r$-graph whose $i$-th part has $m_i$ vertices and does not contain an ordered $K_{s_1,\ldots, s_r}$. In this paper, we show that $$z(m_1,m_2, \cdots, m_{r-1},n ; s_1,s_2, \cdots,s_{r-1}, t)=Θ\left(m_1m_2\cdots m_{r-1} n^{1-1 / s_1s_2\cdots s_{r-1}}\right)$$ for a range of parameters. This extends a result of Conlon [Math. Proc. Camb. Philos. Soc. (2022)].