Further remarks on fractional vs. expectation thresholds

TL;DR

The paper develops an abstract, hypergraph-oriented framework to compare fractional and (fractional) expectation thresholds, building on Talagrand’s conjecture. It generalizes the DeMarco–Kahn approach by formulating sufficient conditions, expressed in terms of overlap parameters $(Y_j)$, under which a random family $G$ of unions from the support of a bounded function $g$ certifies $\langle g\rangle\subseteq\langle G\rangle$ with $w(G,p/L)\le1$. Two key theorems provide concrete criteria guaranteeing the existence of such a $G$ in both overlap-controlled and disjoint-union constructions. The results are then specialized to clique hypergraphs, deriving parameter regimes under which the conjectured threshold relationship holds for hypergraph cliques and yielding a corollary that extends the known clique cases. Overall, the work broadens the toolkit for establishing threshold comparability in structured hypergraphs via probabilistic, union-based constructions.

Abstract

A conjecture of Talagrand (2010) states that the so-called expectation and fractional expectation thresholds are always within at most some constant factor from each other. In this note we generalize a method of DeMarco and Kahn and settle a few more special cases.

Theorems & Definitions (18)

• Conjecture 1: Conjecture 6 from FP23
• Conjecture 2
• Definition 3: General assumptions
• Theorem 4
• Theorem 5
• Lemma 6
• proof : Proof of Lemma \ref{['lem:find_t']}
• Lemma 7
• proof
• proof : Proof of Theorem \ref{['thm:DMK_one']}