Talagrand conjectured that if a family of sets $\mathcal{F}$ over $X = \{ 1,2,\cdots, N \}$ is of large measure, then constant times of unions of sets in $\mathcal{F}$ will cover a large portion of the power set of $X$. This conjecture is a central open problem at the intersection of combinatorics and probability theory, and was described by Talagrand as a personal favorite. This paper provides a proof confirming this conjecture.

翻译：Talagrand 猜想：若定义在 $X = \\{ 1,2,\\cdots, N \\}$ 上的集合族 $\\mathcal{F}$ 具有较大的测度，则 $\\mathcal{F}$ 中集合的常数倍并集将覆盖 $X$ 幂集的很大一部分。该猜想是组合数学与概率论交叉领域的一个核心开放问题，被 Talagrand 本人描述为其个人最喜爱的猜想。本文给出了证实该猜想的证明。