当 $n\ge 2k+3$ 时，集合的相交族通常是平凡的
Intersecting families of sets are usually trivial for $n\ge 2k+3$
AI总结：
本文证明当 $n\ge 2k+3$ 时，几乎所有 $k$ 元均匀相交族都是平凡的，验证了Balogh等人的相关猜想，改进了此前的相交族研究结论。
AI中文摘要：
集合 $[n]$ 的子集族若不含任意两个不相交的集合，则称为相交族；若其所有成员都包含某个公共元素，则称为平凡族。Frankl与Kupavskii，以及Balogh、Das、Liu、Sharifzadeh和Tran分别独立证明，存在常数 $c>0$，当 $n\geq 2k+2+c\sqrt{k\ln k}$ 时，几乎所有 $k$ 元均匀相交族都是平凡的。Balogh、Garcia、Li和Wagner后续将该范围改进为 $n \geq 2k+100\ln k$。本文证明，对于所有 $n\geq 2k+3$，上述结论均成立，从而在该范围内验证了Balogh等人猜想的结论。
英文摘要：
A family of subsets of $[n]$ is called intersecting if it contains no pair of disjoint sets. It is called trivial if all its members contain a common element. Frankl and Kupavskii, and independently Balogh, Das, Liu, Sharifzadeh, and Tran, proved that there is a constant $c>0$ such that, whenever $n \geq 2k+2+c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are trivial. Balogh, Garcia, Li, and Wagner later improved this range to $n \geq 2k+100\ln k$. In this paper, we prove that the same conclusion holds for every $n\geq 2k+3$. This verifies the conjectured conclusion of Balogh, Garcia, Li, and Wagner throughout this range.
