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- W3080453488 abstract "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 $tausubset V$ such that $taunotin X$ but $sigmain X$ for any $sigmasubsetneq tau$. We prove that the $d$-boxicity of a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$ is at most $leftlfloorfrac{1}{d+1}binom{n}{d}rightrfloor$. 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 $frac{1}{d+1}binom{n}{d}$." @default.
- W3080453488 created "2020-09-01" @default.
- W3080453488 creator A5090160306 @default.
- W3080453488 date "2020-08-23" @default.
- W3080453488 modified "2023-09-27" @default.
- W3080453488 title "Representability and boxicity of simplicial complexes" @default.
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