Year: 2025
Type: preprint
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)].
Source: arXiv (Cornell University)
Authors: Gao, Guorong, Hou, Jianfeng, Huang, Shuping, Wang, Hezhi
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2341377552, G3549088517, 图的最大割及其相关问题研究
OpenAlex ID: W7092315381
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