关于（2，3）-合意盒社
On (2,3)-agreeable Box Societies
摘要：
$(k,m)$的概念是由Deborah Berg等人提出的：如果任何大小为$m$的子族包含至少一个非空的$k$ -折叠交集，则$\R^d$的凸子集族被称为$(k,m)$ -可接受的。在那篇文章中，凸族的$(k,m)$ -一致性被证明意味着存在一个具有非空交的尺寸为$βn$的子族，其中$n$是原始族的尺寸，而$β\in[0,1]$是一个只依赖于$k,m$和$d$的显式常数.对于$\R^d$中的$(k,m)$ -合意族，量$β(k,m,d)$被称为最小\emph{agreement proportion}。如果我们只假设集合是凸的，简单的例子表明，$β=0$为$(k,m)$ -同意的家庭在$\R^d$，其中$k<d$。在本文中，我们介绍了新的技术，以找到积极的下界时，限制我们的注意力的家庭$d$盒，即长方体的侧面平行于坐标超平面。我们推导出显式公式的第一个非平凡的情况下：$(2,3)$ -同意家庭的$d$盒与$d\geq 2$的情况下。
英文摘要：
The notion of $(k,m)$-agreeable society was introduced by Deborah Berg et al.: a family of convex subsets of $\R^d$ is called $(k,m)$-agreeable if any subfamily of size $m$ contains at least one non-empty $k$-fold intersection. In that paper, the $(k,m)$-agreeability of a convex family was shown to imply the existence of a subfamily of size $βn$ with non-empty intersection, where $n$ is the size of the original family and $β\in[0,1]$ is an explicit constant depending only on $k,m$ and $d$. The quantity $β(k,m,d)$ is called the minimal \emph{agreement proportion} for a $(k,m)$-agreeable family in $\R^d$. If we only assume that the sets are convex, simple examples show that $β=0$ for $(k,m)$-agreeable families in $\R^d$ where $k<d$. In this paper, we introduce new techniques to find positive lower bounds when restricting our attention to families of $d$-boxes, i.e. cuboids with sides parallel to the coordinates hyperplanes. We derive explicit formulas for the first non-trivial case: the case of $(2,3)$-agreeable families of $d$-boxes with $d\geq 2$.
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