AI 中文总结
该研究确定了随机3-均匀超图中，最大无F子超图是否为二分图的尖锐阈值，这是随机超图Turán型问题的首个尖锐阈值结果。
AI 中文摘要
设F表示法诺平面，即具有7个顶点和7条边的3-均匀超图。Frankl与Füredi，以及独立的Keevash与Sudakov证明，完全3-均匀超图K_n^(3)的最大无F子超图是二分图。本文确定了该性质在随机环境下的尖锐阈值。我们证明，当p̂=Θ_F·n^(-2/3)(log n)^(1/6)（其中Θ_F是依赖于F的显式常数）时，有：(i)若(1+ε)p̂≤p=o(1)，则高概率下，随机3-均匀超图G_{n,p}^{(3)}的每个最大无F子超图都是二分图；(ii)若1/n²≪p≤(1-ε)p̂，则高概率下，G_{n,p}^{(3)}的每个最大无F子超图都不是二分图。据我们所知，这项工作首次给出了随机超图中Turán型问题的尖锐阈值结果。
英文摘要
Let $F$ denote the Fano plane, the $3$-uniform hypergraph with $7$ vertices and $7$ edges. Frankl and Füredi, and independently Keevash and Sudakov, proved that the largest $F$-free subhypergraph of $K_n^{(3)}$ is bipartite. In this paper, we determine the sharp threshold for this property in the random setting. We show that for $\hat{p} = Θ_F \cdot n^{-2/3} \left(\log n\right)^{1/6}$, where $Θ_F$ is an explicit constant depending on $F$, we have: (i) if $(1+ε) \hat{p} \le p = o(1)$, then with high probability every largest $F$-free subhypergraph of $G_{n,p}^{(3)}$ is bipartite; and (ii) if $\frac{1}{n^2} \ll p \le (1-ε) \hat{p}$, then with high probability every largest $F$-free subhypergraph of $G_{n,p}^{(3)}$ is not bipartite. To the best of our knowledge, this work provides the first sharp threshold result obtained for a Turán-type problem in random hypergraphs.
Comments51 pages, 3 figures
