文章基本信息

标题：Bounding Radon Number via Betti Numbers
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作者：patkov:LIPIcs: : , author {Zuzana Pat{'a}kov{'a
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：164
页码：61:1-61:13
DOI：10.4230/LIPIcs.SoCG.2020.61
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We prove general topological Radon-type theorems for sets in â"^d, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly theorem, and consequently the existence of weak Îµ-nets as well as a (p,q)-theorem. More precisely: Let X be either â"^d, smooth real d-manifold, or a finite d-dimensional simplicial complex. Then if F is a finite, intersection-closed family of sets in X such that the ith reduced Betti number (with â"¤â,, coefficients) of any set in F is at most b for every non-negative integer i less or equal to k, then the Radon number of F is bounded in terms of b and X. Here k is the smallest integer larger or equal to d/2 - 1 if X = â"^d; k=d-1 if X is a smooth real d-manifold and not a surface, k=0 if X is a surface and k=d if X is a d-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let F be a finite family of open sets in a surface S such that the intersection of any subfamily of F is either empty, or path-connected. Then the fractional Helly number of F is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a (p,q)-theorem for open subsets of a surface.
关键词：Radon number; topological complexity; constrained chain maps; fractional Helly theorem; convexity spaces