Talagrand’s discrete convexity conjecture

Prove that there exists an integer m such that, for every finite ground set V, every p ∈ (0,1), and every set family F ⊆ 2^V, the implication μ_p(F) ≥ 1 − 1/m entails that F^(m) is p-small, where F^(m) consists of all subsets of V contained in the union of m members of F.

Background

For a set family F and integer m, the paper defines F^m as the family of all sets A ⊆ V for which there exist S_1,...,S_m ∈ F with A ⊆ S_1 ∪ ··* ∪ S_m. Talagrand’s discrete convexity conjecture asserts that sufficiently high product measure of F forces this m-fold union closure to be p-small for some universal integer m.

The paper notes that Chen Li proved a fractional version: if μ_p(F) > 1/2, then F^2 is weakly p/2-small. Consequently, an integral comparison such as Conjecture 1 would supply the remaining step needed for the corresponding integral convexity conclusion. The discrete convexity statement is presented as a conjecture rather than as a result proved in the paper.