Downset thresholds
Abstract
We elucidate the relationship between the threshold and the expectationthreshold of a downset. Qualitatively, our main result demonstrates that there exist downsets with polynomial gaps between their thresholds and expectationthresholds; in particular, the logarithmic gap predictions of KahnKalai and Talagrand (recently proved by ParkPham and FrankstonKahnNarayananPark) about upsets do not apply to downsets. Quantitatively, we show that any collection $\mathcal{G}$ of graphs on $[n]$ that covers the family of all trianglefree 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 bestpossible.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 DOI:
 10.48550/arXiv.2112.08525
 arXiv:
 arXiv:2112.08525
 Bibcode:
 2021arXiv211208525G
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Probability
 EPrint:
 17 pages