License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SOCG.2015.507
URN: urn:nbn:de:0030-drops-51297
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5129/
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Goaoc, Xavier ; Paták, Pavel ; Patáková, Zuzana ; Tancer, Martin ; Wagner, Uli

Bounding Helly Numbers via Betti Numbers

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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.

BibTeX - Entry

@InProceedings{goaoc_et_al:LIPIcs:2015:5129,
  author =	{Xavier Goaoc and Pavel Pat{\'a}k and Zuzana Pat{\'a}kov{\'a} and Martin Tancer and Uli Wagner},
  title =	{{Bounding Helly Numbers via Betti Numbers}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{507--521},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5129},
  URN =		{urn:nbn:de:0030-drops-51297},
  doi =		{10.4230/LIPIcs.SOCG.2015.507},
  annote =	{Keywords: Helly-type theorem, Ramsey’s theorem, Embedding of simplicial complexes, Homological almost-embedding, Betti numbers}
}

Keywords: Helly-type theorem, Ramsey’s theorem, Embedding of simplicial complexes, Homological almost-embedding, Betti numbers
Collection: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 12.06.2015

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