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 $\mathcal{G}$ of graphs on $[n]$ that covers the family of all triangle-free graphs on $[n]$ satisfies the inequality ${\sum }_{G\in \mathcal{G}}\exp (-\delta e(G^c)\mathrm{/}\sqrt{n})<1\mathrm{/}2$ for some universal $\delta >0$, and this is essentially best-possible.