切割超立方体的改进上界
Improved Upper Bounds for Slicing the Hypercube
AI总结：
研究用最少超平面切割n维超立方体所有边的问题，通过构造8个超平面切割Q_{10}，将上界从⌈5n/6⌉改进为⌈4n/5⌉（n为5的奇数倍时加1）。
摘要：
一组超平面$\mathcal{H}$切割顶点集为$\{-1,1\}^n$的$n$维超立方体$Q_n$的所有边，如果对于超立方体中的每条边$e$，存在一个超平面在$\mathcal{H}$中与$e$的内部相交。设$S(n)$为切割$Q_n$所需的最少超平面数。我们证明$S(n) \leq \lceil \frac{4n}{5} \rceil$，除非$n$是5的奇数倍，此时$S(n) \leq \frac{4n}{5} +1$。这改进了Paterson在1971年报告的已知上界$S(n) \leq \lceil\frac{5n}{6} \rceil$。我们还得到了使用$k<n$个超平面可切割的$Q_n$中最大边数的新下界。我们通过构造8个超平面切割$Q_{10}$证明了$S(n)$的改进上界，这得益于最近引入的CPro1：一种自动工具，它使用推理LLM结合自动超参数调优，为发现数学构造创建搜索算法。
英文摘要：
A collection of hyperplanes $\mathcal{H}$ slices all edges of the $n$-dimensional hypercube $Q_n$ with vertex set $\{-1,1\}^n$ if, for every edge $e$ in the hypercube, there exists a hyperplane in $\mathcal{H}$ intersecting $e$ in its interior. Let $S(n)$ be the minimum number of hyperplanes needed to slice $Q_n$. We prove that $S(n) \leq \lceil \frac{4n}{5} \rceil$, except when $n$ is an odd multiple of $5$, in which case $S(n) \leq \frac{4n}{5} +1$. This improves upon the previously known upper bound of $S(n) \leq \lceil\frac{5n}{6} \rceil$ due to Paterson reported in 1971. We also obtain new lower bounds on the maximum number of edges in $Q_n$ that can be sliced using $k<n$ hyperplanes. We prove the improved upper bound on $S(n)$ by constructing $8$ hyperplanes slicing $Q_{10}$ aided by the recently introduced CPro1: an automatic tool that uses reasoning LLMs coupled with automated hyperparameter tuning to create search algorithms for the discovery of mathematical constructions.
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。
