论文标题
简单络合物的能力和拳头性
Representability and boxicity of simplicial complexes
论文作者
论文摘要
让$ x $成为顶点套装$ v $的简单综合体。我们说，如果$ x $是$ d $ - 代表，如果它与$ \ mathbb {r}^d $的凸套装的神经同构。我们将$ d $ boxicity定义为$ x $，为最小$ k $，因此可以写成$ x $作为$ k $ $ d $ d $的交集。这概括了罗伯茨（Roberts）定义的图表的拳头概念。 $ x $的缺失面是$τ\ subset v $，因此对于任何$σ\ subsetneqτ$而言，$τ\ notin x $，但$点平。我们证明，$ n $顶点上的简单复合物的$ d $ - 盒子，而没有丢失大于$ d $的面孔的面孔，最多是$ \ weft \ lfloor \ lfrac {1} {d+1} {d+1} \ binom {n} {n} {d} {d} {d} {d} \ right \ rfloor $。界限很尖锐：简单复合体的$ D $ - 盒子，其缺失的面孔形成了steiner $（d，d+1，n）$ - 系统正好是$ \ frac {1} {d+1} {d+1} \ binom {n} {d} {d} $。
Let $X$ be a simplicial complex on vertex set $V$. We say that $X$ is $d$-representable if it is isomorphic to the nerve of a family of convex sets in $\mathbb{R}^d$. We define the $d$-boxicity of $X$ as the minimal $k$ such that $X$ can be written as the intersection of $k$ $d$-representable simplicial complexes. This generalizes the notion of boxicity of a graph, defined by Roberts. A missing face of $X$ is a set $τ\subset V$ such that $τ\notin X$ but $σ\in X$ for any $σ\subsetneq τ$. We prove that the $d$-boxicity of a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$ is at most $\left\lfloor\frac{1}{d+1}\binom{n}{d}\right\rfloor$. The bound is sharp: the $d$-boxicity of a simplicial complex whose set of missing faces form a Steiner $(d,d+1,n)$-system is exactly $\frac{1}{d+1}\binom{n}{d}$.
