Affiliation:
1. Department of Mathematics Rutgers University Piscataway New Jersey USA
2. Department of Mathematics Princeton University Princeton New Jersey USA
Abstract
AbstractWe 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 of graphs on that covers the family of all triangle‐free graphs on satisfies the inequality for some universal , and this is essentially best‐possible.
Funder
National Science Foundation
Subject
Applied Mathematics,Computer Graphics and Computer-Aided Design,General Mathematics,Software
Cited by
1 articles.
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1. Note on down‐set thresholds;Random Structures & Algorithms;2023-11-22