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Bounding Helly numbers via Betti numbers

The result's identifiers

  • Result code in IS VaVaI

    <a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F15%3A10312213" target="_blank" >RIV/00216208:11320/15:10312213 - isvavai.cz</a>

  • Result on the web

    <a href="http://drops.dagstuhl.de/opus/volltexte/2015/5129/" target="_blank" >http://drops.dagstuhl.de/opus/volltexte/2015/5129/</a>

  • DOI - Digital Object Identifier

    <a href="http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.507" target="_blank" >10.4230/LIPIcs.SOCG.2015.507</a>

Alternative languages

  • Result language

    angličtina

  • Original language name

    Bounding Helly numbers via Betti numbers

  • Original language description

    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 numberat 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.

  • Czech name

  • Czech description

Classification

  • Type

    D - Article in proceedings

  • CEP classification

    BA - General mathematics

  • OECD FORD branch

Result continuities

  • Project

    <a href="/en/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Center of excellence - Institute for theoretical computer science (CE-ITI)</a><br>

  • Continuities

    S - Specificky vyzkum na vysokych skolach<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Others

  • Publication year

    2015

  • Confidentiality

    S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Data specific for result type

  • Article name in the collection

    Proceedings of the 31st International Symposium on Computational Geometry (SoCG 2015)

  • ISBN

    978-3-939897-83-5

  • ISSN

  • e-ISSN

  • Number of pages

    15

  • Pages from-to

    507-521

  • Publisher name

    LIPICS

  • Place of publication

    Dagstuhl

  • Event location

    Eindhoven

  • Event date

    Jun 22, 2015

  • Type of event by nationality

    WRD - Celosvětová akce

  • UT code for WoS article