A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl–Kupavskii and Balogh–Das–Liu–Sharifzadeh–Tran showed that for $n\ge 2k+c\sqrt{k\ln k}$, almost all k-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for $n\ge 2k+100\ln k$. Our proof uses, among others, the graph container method and the Das–Tran removal lemma.

中文翻译：

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集合的相交族通常是微不足道的

一个家族的子集$[n]$如果它的每对集合都相交，则它是相交的。确定大相交族的结构是极值组合学的核心问题。Frankl–Kupavskii 和 Balogh–Das–Liu–Sharifzadeh–Tran 证明了对于$n\ge 2k+C\sqrt{k\mathrm{因}k}$，几乎所有k均匀相交族都是星形。改进他们的结果，我们表明同样的结论适用于$n\ge 2k+100\mathrm{因}k$。我们的证明使用了图容器方法和 Das-Tran 去除引理等。