On the "second" Kahn--Kalai Conjecture

TL;DR

This paper advances understanding of the threshold for random graphs to contain a specific subgraph, providing bounds involving logarithmic factors and connecting to recent theorems on fractional thresholds.

Contribution

It establishes new bounds on the fractional expectation threshold related to the second Kahn-Kalai Conjecture, refining previous estimates and linking to recent fractional relaxation results.

Findings

Proves $p_{ ext{E}}^*(H)=O(p_{ ext{E}}(H) ext{log}^2 n)$

Shows $p_c(H)=O(p_{ ext{E}}(H) ext{log}^3 n)$ using recent theorems

Identifies potential improvements if logarithmic factors can be removed

Abstract

We make progress on a conjecture of Kahn and Kalai, the original (stronger but less general) version of what became known as the ``Kahn-Kalai Conjecture" (KKC; now a theorem of Park and Pham). This ``second" KKC concerns the threshold, $p_c(H)$, for $G_{n,p}$ to contain a copy of a given graph $H$, predicting $p_c(H) = O(p_{\mathbb E}(H)\log n)$, where $p_{\mathbb E}$ is an easy lower bound on $p_c$. What we actually show is $p_{\mathbb E}^*(H)=O(p_{\mathbb E}(H)\log ^2n)$, where $p_{\mathbb E}^*$, the fractional expectation threshold, is a larger lower bound suggested by Talagrand. When combined with Talagrand's fractional relaxation of the KKC (now a theorem of Frankston, Kahn, Narayanan and Park), this gives $p_c(H)=O(p_{\mathbb E}(H)\log^3 n)$. (The second KKC would follow similarly if one could remove the log factors from the above bound on $p_{\mathbb E}^*$.)