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Hypergraphs without complete partite subgraphs
- Part of
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- Journal:
- Combinatorics, Probability and Computing , First View
- Published online by Cambridge University Press:
- 11 December 2025, pp. 1-4
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Fix integers
$r \ge 2$ and 
$1\le s_1\le \cdots \le s_{r-1}\le t$ and set 
$s=\prod _{i=1}^{r-1}s_i$. Let 
$K=K(s_1, \ldots , s_{r-1}, t)$ denote the complete 
$r$-partite 
$r$-uniform hypergraph with parts of size 
$s_1, \ldots , s_{r-1}, t$. We prove that the Zarankiewicz number 
$z(n, K)= n^{r-1/s-o(1)}$ provided 
$t\gt 3^{s+o(s)}$. Previously this was known only for 
$t \gt ((r-1)(s-1))!$ 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 
$s_i$, for example, it gives 
$z(n, K(2,2,7))=n^{11/4-o(1)}$ where the exponent 11/4 is optimal, whereas previously this was only known with 7 replaced by 721.
