---
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@
...