---
res:
bibo_abstract:
- 'We show that very weak topological assumptions are enough to ensure the existence
of a Helly-type theorem. More precisely, we show that for any non-negative integers
b and d there exists an integer h(b,d) such that the following holds. If F is
a finite family of subsets of R^d such that the ith reduced Betti number (with
Z_2 coefficients in singular homology) of the intersection of any proper subfamily
G of F is at most b for every non-negative integer i less or equal to (d-1)/2,
then F has Helly number at most h(b,d). These topological conditions are sharp:
not controlling any of these first Betti numbers allow for families with unbounded
Helly number. Our proofs combine homological non-embeddability results with a
Ramsey-based approach to build, given an arbitrary simplicial complex K, some
well-behaved chain map from C_*(K) to C_*(R^d). Both techniques are of independent
interest.@eng'
bibo_authorlist:
- foaf_Person:
foaf_givenName: Xavier
foaf_name: Goaoc, Xavier
foaf_surname: Goaoc
- foaf_Person:
foaf_givenName: Pavel
foaf_name: Paták, Pavel
foaf_surname: Paták
- foaf_Person:
foaf_givenName: Zuzana
foaf_name: Patakova, Zuzana
foaf_surname: Patakova
orcid: 0000-0002-3975-1683
- foaf_Person:
foaf_givenName: Martin
foaf_name: Tancer, Martin
foaf_surname: Tancer
orcid: 0000-0002-1191-6714
- foaf_Person:
foaf_givenName: Uli
foaf_name: Wagner, Uli
foaf_surname: Wagner
foaf_workInfoHomepage: http://www.librecat.org/personId=36690CA2-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-1494-0568
bibo_doi: 10.4230/LIPIcs.SOCG.2015.507
bibo_volume: 34
dct_date: 2015^xs_gYear
dct_language: eng
dct_publisher: ACM@
dct_title: Bounding Helly numbers via Betti numbers@
...