标题： 超图的边统计猜想 显示英文标题

标题： The edge-statistics conjecture for hypergraphs

摘要： 设$r,k,\ell$为整数，使得$0\le\ell\le\binom{k}{r}$。 给定一个大的$r$-一致超图$G$，我们考虑恰好包含$\ell$条边的$k$-顶点子集的比例。 如果$\ell$为 0 或$\binom{k}{r}$，这个分数可以精确为 1 (通过将$G$取为空或完整)，但对于$\ell$的所有其他值，人们可能会怀疑这个分数总是明显小于 1。 在本文中，我们证明了一个本质上最优的结果：如果$\ell$不是 0 或者$\binom{k}{r}$，那么这个分数最多为$(1/e) + \varepsilon$，假设$k$在关于$r$的足够大，并且$\varepsilon>0$足够大，而$G$在关于$k$的足够大。 先前，这仅在$r,k,\ell$的非常有限的取值范围内被知晓（由于 Kwan-Sudakov-Tran、Fox-Sauermann 和 Martinsson-Mousset-Noever-Trujić）。我们的结果回答了 Alon-Hefetz-Krivelevich-Tyomkyn 的一个问题，他们将此视为其“边统计猜想”的超图推广。我们还证明了一个更强的界限，当$\ell$远离 0 和$\binom{k}{r}$时。 显示英文摘要

摘要： Let $r,k,\ell$ be integers such that $0\le\ell\le\binom{k}{r}$. Given a large $r$-uniform hypergraph $G$, we consider the fraction of $k$-vertex subsets which span exactly $\ell$ edges. If $\ell$ is 0 or $\binom{k}{r}$, this fraction can be exactly 1 (by taking $G$ to be empty or complete), but for all other values of $\ell$, one might suspect that this fraction is always significantly smaller than 1. In this paper we prove an essentially optimal result along these lines: if $\ell$ is not 0 or $\binom{k}{r}$, then this fraction is at most $(1/e) + \varepsilon$, assuming $k$ is sufficiently large in terms of $r$ and $\varepsilon>0$, and $G$ is sufficiently large in terms of $k$. Previously, this was only known for a very limited range of values of $r,k,\ell$ (due to Kwan-Sudakov-Tran, Fox-Sauermann, and Martinsson-Mousset-Noever-Truji\'{c}). Our result answers a question of Alon-Hefetz-Krivelevich-Tyomkyn, who suggested this as a hypergraph generalisation of their "edge-statistics conjecture". We also prove a much stronger bound when $\ell$ is far from 0 and $\binom{k}{r}$.