Note on down‐set thresholds

Abstract

AbstractGunby–He–Narayanan showed that the logarithmic gap predictions of Kahn–Kalai and Talagrand (proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about thresholds of up‐sets do not apply to down‐sets. In particular, for the down‐set of triangle‐free graphs, they showed that there is a polynomial gap between the threshold and the factional expectation threshold. In this short note we give a simpler proof of this result, and extend the polynomial threshold gap to down‐sets of ‐free graphs.