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Bounding Helly Numbers via Betti Numbers

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.
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Contributor : Xavier Goaoc Connect in order to contact the contributor
Submitted on : Monday, August 28, 2017 - 12:58:19 PM
Last modification on : Wednesday, November 10, 2021 - 5:38:05 PM

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Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, Uli Wagner. Bounding Helly Numbers via Betti Numbers. International Symposium on Computational Geometry, 2015, Eindhoven, Netherlands. pp.507--521, ⟨10.4230/LIPIcs.SOCG.2015.507⟩. ⟨hal-01577897⟩



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