When do the Kahn-Kalai Bounds Provide Nontrivial Information?

TL;DR

This work studies when the Park-Pham (Kahn-Kalai) bounds yield information beyond the trivial bound $p_c(\mathcal{F})<1$ for nontrivial upper sets $\mathcal{F}\subseteq 2^X$. It analyzes the interaction between the expectation threshold $q(\mathcal{F})$ and the size of the largest minimal element $\ell(\mathcal{F})$, showing that asymptotically perfect information occurs when $\log \ell(\mathcal{F}_n) \ll 1/ q(\mathcal{F}_n)$. In the regime $\ell(\mathcal{F}_n) \to \infty$, it derives strong necessary conditions: the number of minimal elements must grow without bound, and for every fixed $t$, the intersection of all-but-$t$ minimal elements must be empty for all large $n$, which implies a wedge-like spread in $2^{X_n}$. These results delineate the structural situations in which the Park-Pham bounds provide genuinely new information about thresholds and relate the bounds to the combinatorial geometry of the family.

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.

Theorems & Definitions (24)

• Remark 1.1
• Remark 1.2
• Definition 2.1: Threshold
• Theorem 2.2: Park-Pham Theorem
• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Theorem 4.1
• proof
• Lemma 4.2