On a problem of M. Talagrand

We address a special case of a conjecture of M. Talagrand relating two notions of “threshold” for an increasing family F of subsets of a finite set V . The full conjecture implies equivalence of the “Fractional Expectation-Threshold Conjecture,” due to Talagrand and recently proved by the authors and B. Narayanan, and the (stronger) “Expectation-Threshold Conjecture” of the second author and G. Kalai. The conjecture under discussion here says there is a fixed L such that if, for a given F , p ∈ [0, 1] admits λ : 2V → R+ with ∑ S⊆F λS ≥ 1 ∀F ∈ F and ∑ S λSp |S| ≤ 1/2 (a.k.a. F is weakly p-small), then p/L admits such a λ taking values in {0, 1} (F is (p/L)-small). Talagrand showed this when λ is supported on singletons and suggested, as a more challenging test case, proving it when λ is supported on pairs. The present work provides such a proof.