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arXiv · 2606.12133

On a hypergraph Tur\'an problem of Balogh-Bohman-Bollob\'as-Zhao

Heng Li·Jie Ma·Tianhen Wang·Yixiao Zhang·Tianming Zhu
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Abstract

Let $S$ and $T$ be disjoint sets with $|S|=i$ and $|T|=r-1$ for $2\le i\le r-1$, and let $B_i^{(r)}$ be the $r$-graph on $S\cup T$ whose edges are the $r$-subsets containing $S$ or $T$. We study the deficit $q_{r,i}:=1-\pi(B_i^{(r)})$ in its Tur\'an density. Balogh, Bohman, Bollob\'as, and Zhao previously obtained bounds for these deficits with logarithmic gaps near both ends of the sequence $B_i^{(r)}$, namely, when $i=O(1)$ or $i=r-O(1)$. We close these gaps by showing that, as $r\to\infty$, for every fixed integer $a\ge1$, $q_{r,a+1}=\Theta_a(r^{-a})$, and for every fixed integer $b\ge2$, $q_{r,r-b}=\Theta_b(r^{-b}\log r)$.

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Heng Li, Jie Ma, Tianhen Wang, Yixiao Zhang, Tianming Zhu. 2026-06-10. On a hypergraph Tur\'an problem of Balogh-Bohman-Bollob\'as-Zhao. https://arxiv.org/abs/2606.12133

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