arXiv 2209.033262022-09-07math.COcs.DMmath.PR

关于第二个Kahn-Kalai猜想

On the Second Kahn--Kalai Conjecture

Elchanan Mossel, Jonathan Niles-Weed, Nike Sun, Ilias Zadik

备注：: 4 pages

摘要：

对于任何给定的图$H$，我们感兴趣的是$p_\mathrm{crit}(H)$，最小的$p$，使得Erdens-Rényi图$G(n,p)$包含一个概率至少为$1/2$的$H$的副本。Kahn和Kalai（2007）指出，$p_\mathrm{crit}(H)$被一个更简单的“子图期望阈值”$p_\mathrm{E}(H)$赋予对数因子，$p_\mathrm{E}(H)$是最小的$p$，使得对于每个子图$H'\subseteq H$，Erdens-Rényi图$G(n,p)$至少包含$H'$的\emph{in expectation} $1/2$拷贝。$p_\mathrm{E}(H) \le p_\mathrm{crit}(H)$是平凡的，所谓的“第二Kahn-Kalai猜想”指出$p_\mathrm{crit}(H) \lesssim p_\mathrm{E}(H) \log e(H)$其中$e(H)$是$H$中的边数。在这篇文章中，我们提出了一个自然的修改$p_\mathrm{E, new}(H)$的Kahn-Kalai子图的期望阈值，我们表明是夹在$p_\mathrm{E}(H)$和$p_\mathrm{crit}(H)$。新定义$p_\mathrm{E, new}(H)$基于以下简单观察：如果$G(n,p)$包含$H$的拷贝，而$H$包含$H'$的\emph{many}拷贝，则$G(n,p)$也必须包含$H'$的\emph{many}拷贝。然后，我们表明，$p_\mathrm{crit}(H) \lesssim p_\mathrm{E, new}(H) \log e(H)$，从而证明了第二个Kahn-Kalai猜想的修改。该界限如下的直接应用集理论的“蔓延”的财产，这导致了最近突破的向日葵猜想由Alweiss，Lovett，吴和张和第一个分数卡恩-卡莱猜想由Frankston，卡恩，纳拉亚南和公园。

英文摘要：

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.

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