Down-set thresholds
Benjamin Gunby, Xiaoyu He, and Bhargav Narayanan

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 of graphs on that covers the family of all triangle-free graphs on satisfies the inequality for some universal , and this is essentially best-possible.
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