Fractional Vs. Expectation Thresholds: Random Support Case

TL;DR

This paper proves that for random hypergraphs, the expectation and fractional expectation thresholds are almost surely within a constant factor of each other, confirming a conjecture by Talagrand in the unweighted case.

Contribution

It establishes the asymptotic equivalence of these thresholds for random hypergraphs, advancing understanding of threshold phenomena in probabilistic combinatorics.

Findings

Expectation and fractional expectation thresholds are within a constant factor in random hypergraphs.

The result confirms Talagrand's conjecture for the unweighted case.

Thresholds are asymptotically equivalent in the random support setting.

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. We prove for the unweighted case that this is a.a.s. true when the support is a random hypergraph.