A Proof of the Kahn-Kalai Conjecture

TL;DR

This paper proves the Kahn-Kalai expectation-threshold conjecture, establishing a bound relating the threshold and expectation threshold of increasing properties on finite sets, with implications for probabilistic combinatorics.

Contribution

It provides a proof of the long-standing conjecture, connecting threshold concepts with a logarithmic factor based on minimal member size.

Findings

Proves the expectation-threshold conjecture for increasing properties.

Establishes a bound: $p_c() = O(q() \, \log \ell(\f))$.

Advances understanding of phase transitions in combinatorial structures.

Abstract

Proving the ``expectation-threshold'' conjecture of Kahn and Kalai, we show that for any increasing property $\mathcal{F}$ on a finite set $X$, $$p_c(\mathcal{F})=O(q(\mathcal{F})\log \ell(\mathcal{F})),$$ where $p_c(\mathcal{F})$ and $q(\mathcal{F})$ are the threshold and ``expectation threshold'' of $\mathcal{F}$, and $\ell(\mathcal{F})$ is the maximum of $2$ and the maximum size of a minimal member of $\mathcal{F}$.