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Hypergraphs without complete partite subgraphs

Combinatorics 2026-05-20 v2

Abstract

Fix integers r2r \ge 2 and 1s1sr1t1\le s_1\le \cdots \le s_{r-1}\le t and set s=i=1r1sis=\prod_{i=1}^{r-1}s_i. Let K=K(s1,,sr1,t)K=K(s_1, \ldots, s_{r-1}, t) denote the complete rr-partite rr-uniform hypergraph with parts of size s1,,sr1,ts_1, \ldots, s_{r-1}, t. We prove that the Zarankiewicz number z(n,K)=nr1/so(1)z(n, K)= n^{r-1/s-o(1)} provided t>3s+o(s)t> 3^{s+o(s)}. Previously this was known only for t>((r1)(s1))!t > ((r-1)(s-1))! due to Pohoata and Zakharov. Our novel approach, which uses Behrend's construction of sets with no 3 term arithmetic progression, also applies for small values of sis_i, for example, it gives z(n,K(2,2,7))=n11/4o(1)z(n, K(2,2,7))=n^{11/4-o(1)} where the exponent 11/4 is optimal, whereas previously this was only known with 7 replaced by 721.

Keywords

Cite

@article{arxiv.2507.06390,
  title  = {Hypergraphs without complete partite subgraphs},
  author = {Dhruv Mubayi},
  journal= {arXiv preprint arXiv:2507.06390},
  year   = {2026}
}

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6 pages