--- res: bibo_abstract: - '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.@eng' bibo_authorlist: - foaf_Person: foaf_givenName: Zuzana foaf_name: Patakova, Zuzana foaf_surname: Patakova foaf_workInfoHomepage: http://www.librecat.org/personId=48B57058-F248-11E8-B48F-1D18A9856A87 orcid: 0000-0002-3975-1683 bibo_doi: 10.4230/LIPIcs.SoCG.2020.61 bibo_volume: 164 dct_date: 2020^xs_gYear dct_isPartOf: - http://id.crossref.org/issn/18688969 - http://id.crossref.org/issn/9783959771436 dct_language: eng dct_publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik@ dct_title: Bounding radon number via Betti numbers@ ...