Down‐set thresholds

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 𝒢 of graphs on [n] that covers the family of all triangle‐free graphs on [n] satisfies the inequality ∑G∈𝒢exp(−δe(Gc)/n)<1/2 for some universal δ>0 , and this is essentially best‐possible.