On the Second Kahn--Kalai Conjecture

Abstract

For any given graph $H$, we are interested in $p_\mathrm{crit}(H)$, the minimal $p$ such that the Erdős-Rényi graph $G(n,p)$ contains a copy of $H$ with probability at least $1/2$. Kahn and Kalai (2007) conjectured that $p_\mathrm{crit}(H)$ is given up to a logarithmic factor by a simpler "subgraph expectation threshold" $p_\mathrm{E}(H)$, which is the minimal $p$ such that for every subgraph $H'\subseteq H$, the Erdős-Rényi graph $G(n,p)$ contains \emph{in expectation} at least $1/2$ copies of $H'$. It is trivial that $p_\mathrm{E}(H) \le p_\mathrm{crit}(H)$, and the so-called "second Kahn-Kalai conjecture" states that $p_\mathrm{crit}(H) \lesssim p_\mathrm{E}(H) \log e(H)$ where $e(H)$ is the number of edges in $H$. In this article, we present a natural modification $p_\mathrm{E, new}(H)$ of the Kahn--Kalai subgraph expectation threshold, which we show is sandwiched between $p_\mathrm{E}(H)$ and $p_\mathrm{crit}(H)$. The new definition $p_\mathrm{E, new}(H)$ is based on the simple observation that if $G(n,p)$ contains a copy of $H$ and $H$ contains \emph{many} copies of $H'$, then $G(n,p)$ must also contain \emph{many} copies of $H'$. We then show that $p_\mathrm{crit}(H) \lesssim p_\mathrm{E, new}(H) \log e(H)$, thus proving a modification of the second Kahn--Kalai conjecture. The bound follows by a direct application of the set-theoretic "spread" property, which led to recent breakthroughs in the sunflower conjecture by Alweiss, Lovett, Wu and Zhang and the first fractional Kahn--Kalai conjecture by Frankston, Kahn, Narayanan and Park.