A 0/1-polytope is the convex hull of a subset $V\subseteq \{0,1\}^n$. A celebrated conjecture of Mihail and Vazirani asserts that the graph of every 0/1-polytope has edge-expansion at least 1. In this paper, we show that typical 0/1-polytopes have significantly stronger expansion. Specifically, if $V$ is formed by sampling each vertex of $\{0,1\}^n$ independently with constant probability $p$, then with high probability the edge-expansion is $Θ(n)$ for $p \in (1/2, 1)$, and $n^{Θ(\log \log n)}$ for $p \in (0, 1/2)$. This improves the previously best known bound $Ω(1)$ due to Ferber, Krivelevich, Sales and Samotij.

翻译：0/1多面体是子集 $V\subseteq \{0,1\}^n$ 的凸包。Mihail和Vazirani的一个著名猜想断言，每个0/1多面体的图的边扩张至少为1。在本文中，我们表明典型的0/1多面体具有显著更强的扩张性质。具体而言，若 $V$ 是通过以常数概率 $p$ 独立采样 $\{0,1\}^n$ 的每个顶点形成的，则对于 $p \in (1/2, 1)$，边扩张以高概率为 $Θ(n)$，而对于 $p \in (0, 1/2)$，边扩张为 $n^{Θ(\log \log n)}$。这改进了此前由Ferber、Krivelevich、Sales和Samotij证明的最佳已知界 $Ω(1)$。