Bounding Radon Number via Betti Numbers
We prove general topological Radon-type theorems for sets in ℝ^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 ε-nets as well as a (p,q)-theorem.
More precisely: Let X be either ℝ^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 ℤ₂ 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 = ℝ^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.
Radon number
topological complexity
constrained chain maps
fractional Helly theorem
convexity spaces
Mathematics of computing~Geometric topology
Theory of computation~Computational geometry
61:1-61:13
Regular Paper
The research stay of the author at IST Austria is funded by the project Improvement of internationalization in the field of research and development at Charles University, through the support of quality projects MSCA-IF (CZ.02.2.69/0.0/0.0/17_050/0008466).
A full version of the paper is available at https://arxiv.org/abs/1908.01677.
First and foremost, I am very grateful to Pavel Paták for numerous discussions, helpful suggestions and a proofreading. Many thanks to Xavier Goaoc for his feedback and comments, which have been very helpful in improving the overall presentation. I also thank Endre Makai for pointers to relevant literature, especially to the book [Soltan, 1984]. Finally, many thanks to Natan Rubin for several discussions at the very beginning of the project.
Zuzana
Patáková
Zuzana Patáková
Computer Science Institute, Charles University, Prague, Czech Republic
IST Austria, Klosterneuburg, Austria
https://orcid.org/0000-0002-3975-1683
10.4230/LIPIcs.SoCG.2020.61
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Zuzana Patáková
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