边统计量猜想证明的一个补充
A Completion of the Proof of the Edge-statistics Conjecture
摘要：
对于给定的整数$k$和$\ell$与$0<\ell< {k \choose 2}$，Alon，Hefetz，Krivelevich和Tyomkyn提出了如下猜想：当从一个非常大的图$G$中随机均匀地抽样一个$k$ -顶点子集时，在抽样的$k$ -顶点子集中恰好有$\ell$边的概率至多为$e^{-1}+o_k(1)$ . Kwan，Sudakov和Tran在$Ω(k)\leq \ell\leq {k \choose 2}-Ω(k)$的例子中证明了这个猜想.在本文中，我们通过解决剩余的情况完成了证明的猜想。此外，我们给出了几乎紧的上界的情况下，上述概率$ω(1)\leq \ell\leq o(k)$。我们也将我们的一些结果推广到了边有界的超图。
英文摘要：
For given integers $k$ and $\ell$ with $0<\ell< {k \choose 2}$, Alon, Hefetz, Krivelevich and Tyomkyn formulated the following conjecture: When sampling a $k$-vertex subset uniformly at random from a very large graph $G$, then the probability to have exactly $\ell$ edges within the sampled $k$-vertex subset is at most $e^{-1}+o_k(1)$. This conjecture was proved in the case $Ω(k)\leq \ell\leq {k \choose 2}-Ω(k)$ by Kwan, Sudakov and Tran. In this paper, we complete the proof of the conjecture by resolving the remaining cases. We furthermore give nearly tight upper bounds for the probability described above in the case $ω(1)\leq \ell\leq o(k)$. We also extend some of our results to hypergraphs with bounded edge size.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
