A Proof of Talagrand's Creating Large Sets Conjecture

TL;DR

This paper proves Talagrand's conjecture that large measure families of sets, when combined through unions, cover a significant portion of the power set, advancing understanding in combinatorics and probability.

Contribution

The paper provides the first proof confirming Talagrand's long-standing conjecture on the coverage of large measure set families.

Findings

Confirmed Talagrand's conjecture

Established bounds on unions of large measure sets

Enhanced understanding of set families in combinatorics

Abstract

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.