Down-set thresholds

TL;DR

This paper explores the relationship between thresholds and expectation-thresholds of down-sets, revealing polynomial gaps and disproving certain logarithmic gap predictions for down-sets, with implications in graph theory.

Contribution

It demonstrates the existence of polynomial gaps between thresholds and expectation-thresholds for down-sets and provides bounds for graph collections covering triangle-free graphs.

Findings

Existence of polynomial gaps between thresholds and expectation-thresholds for down-sets.

Disproof of logarithmic gap predictions for down-sets.

A bound on sums over graph collections covering triangle-free graphs.

Abstract

We elucidate the relationship between the threshold and the expectation-threshold of a down-set. Qualitatively, our main result demonstrates that there exist down-sets with polynomial gaps between their thresholds and expectation-thresholds; in particular, the logarithmic gap predictions of Kahn--Kalai and Talagrand (recently proved by Park--Pham and Frankston--Kahn--Narayanan--Park) about up-sets do not apply to down-sets. Quantitatively, we show that any collection $\mathcal{G}$ of graphs on $[n]$ that covers the family of all triangle-free graphs on $[n]$ satisfies the inequality $\sum_{G \in \mathcal{G}} \exp(-\delta e(G^c) / \sqrt{n}) < 1/2$ for some universal $\delta > 0$, and this is essentially best-possible.