eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-08
61:1
61:13
10.4230/LIPIcs.SoCG.2020.61
article
Bounding Radon Number via Betti Numbers
Patáková, Zuzana
1
2
https://orcid.org/0000-0002-3975-1683
Computer Science Institute, Charles University, Prague, Czech Republic
IST Austria, Klosterneuburg, Austria
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol164-socg2020/LIPIcs.SoCG.2020.61/LIPIcs.SoCG.2020.61.pdf
Radon number
topological complexity
constrained chain maps
fractional Helly theorem
convexity spaces