Abstract
We prove general topological Radontype 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 εnets as well as a (p,q)theorem.
More precisely: Let X be either ℝ^d, smooth real dmanifold, or a finite ddimensional simplicial complex. Then if F is a finite, intersectionclosed 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 nonnegative 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=d1 if X is a smooth real dmanifold and not a surface, k=0 if X is a surface and k=d if X is a ddimensional 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 pathconnected. 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.
BibTeX  Entry
@InProceedings{patkov:LIPIcs:2020:12219,
author = {Zuzana Pat{\'a}kov{\'a}},
title = {{Bounding Radon Number via Betti Numbers}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {61:161:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771436},
ISSN = {18688969},
year = {2020},
volume = {164},
editor = {Sergio Cabello and Danny Z. Chen},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12219},
URN = {urn:nbn:de:0030drops122198},
doi = {10.4230/LIPIcs.SoCG.2020.61},
annote = {Keywords: Radon number, topological complexity, constrained chain maps, fractional Helly theorem, convexity spaces}
}
Keywords: 

Radon number, topological complexity, constrained chain maps, fractional Helly theorem, convexity spaces 
Collection: 

36th International Symposium on Computational Geometry (SoCG 2020) 
Issue Date: 

2020 
Date of publication: 

08.06.2020 