On the Maximum $F_5$-Free Subhypergraphs of a Random Hypergraph

Pub Date : 2023-11-03 DOI:10.37236/11328
Igor Araujo, József Balogh, Haoran Luo
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Abstract

Denote by $F_5$ the $3$-uniform hypergraph on vertex set $\{1,2,3,4,5\}$ with hyperedges $\{123,124,345\}$. Balogh, Butterfield, Hu, and Lenz proved that if $p > K \log n /n$ for some large constant $K$, then every maximum $F_5$-free subhypergraph of $G^3(n,p)$ is tripartite with high probability, and showed that if $p_0 = 0.1\sqrt{\log n} /n$, then with high probability there exists a maximum $F_5$-free subhypergraph of $G^3(n,p_0)$ that is not tripartite. In this paper, we sharpen the upper bound to be best possible up to a constant factor. We prove that if $p > C \sqrt{\log n} /n $ for some large constant $C$, then every maximum $F_5$-free subhypergraph of $G^3(n, p)$ is tripartite with high probability.
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关于随机超图的最大$F_5$ free子超图
用$F_5$表示顶点集$\{1,2,3,4,5\}$上具有超边$\{123,124,345\}$的$3$ -一致超图。Balogh, Butterfield, Hu, and Lenz证明了如果$p > K \log n /n$对于某大常数$K$,那么$G^3(n,p)$的每一个极大$F_5$自由子超图都有大概率是三方的,并且证明了如果$p_0 = 0.1\sqrt{\log n} /n$,那么有大概率存在$G^3(n,p_0)$的极大$F_5$自由子超图不是三方的。在本文中,我们锐化上界,使其尽可能达到一个常数因子。证明了如果$p > C \sqrt{\log n} /n $对于某大常数$C$,则$G^3(n, p)$的每一个极大$F_5$自由子超图都是高概率的三部。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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