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arXiv · 2112.08525

Down-set thresholds

Benjamin Gunby·Xiaoyu He·Bhargav Narayanan
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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 $\mathcal{G}$ of graphs on $[n]$ that covers the family of all triangle-free graphs on $[n]$ satisfies the inequality $\sum_{G \in \mathcal{G}} \exp(-δe(G^c) / \sqrt{n}) < 1/2$ for some universal $δ> 0$, and this is essentially best-possible.

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Benjamin Gunby, Xiaoyu He, Bhargav Narayanan. 2023-02-02. Down-set thresholds. https://arxiv.org/abs/2112.08525

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