When do the Kahn-Kalai Bounds Provide Nontrivial Information?

TL;DR

This paper investigates when the Kahn-Kalai bounds on critical probabilities in property thresholds provide meaningful information, establishing necessary and sufficient conditions for their nontriviality in large-scale combinatorial structures.

Contribution

It offers new necessary and sufficient conditions for the Kahn-Kalai bounds to be nontrivially informative in the asymptotic regime.

Findings

Necessary condition: minimal elements' collections must have finitely many nonempty intersections.

Sufficient conditions identified for bounds to be asymptotically perfect.

Sequences must occupy an expanding wedge in the power set, spreading across the structure.

Abstract

The Park-Pham theorem (previously known as the Kahn-Kalai conjecture), bounds the critical probability, $p_c(\mathcal{F})$, of the a non-trivial property $\mathcal{F}\subseteq 2^X$ that is closed under supersets by the product of a universal constant $K$, the expectation threshold of the property, $q(\mathcal{F})$, and the logarithm of the size of the property's largest minimal element, $\log\ell(\mathcal{F})$. That is, the Park-Pham theorem asserts that $p_c(\mathcal{F})\leq Kq(\mathcal{F})\log\ell(\mathcal{F})$. Since the critical probability $p_c(\mathcal{F})$ always satisfies $p_c(\mathcal{F})<1$, one may ask when the upper bound posed by Kahn and Kalai gives us more information than this--that is, when is it true that $Kq(\mathcal{F})\log\ell(\mathcal{F}) < 1$? In this short note, we provide a number of necessary conditions for this to happen and give a few sufficient conditions for the bounds to provide new (and, in fact, asymptotically perfect) information along the way. In the most interesting case where $\ell(\mathcal{F}_n)\rightarrow \infty$, we prove the following relatively strong necessary condition for the Kahn-Kalai bounds to provide nontrivial information: For every positive integer $t$, every collection of all-but-$t$ of the minimal elements of $\mathcal{F}_n$ may have nonempty intersection for only finitely many $n$. Consequently, not only must the number of minimal elements become arbitrarily large, but so too must the size of any cover. Intuitively, this means that such sequences $\mathcal{F}_n$ must occupy an ever-widening `wedge' in $2^{X_n}$: the further $\mathcal{F}_n$ climbs up $2^{X_n}$ in one area, the further it must spread down and across $2^{X_n}$ in another.