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 Rd such that βi(∩G)≤b for any G⊊F and every 0 ≤ i ≤ [d/2]-1 then F has Helly number at most h(b, d). Here βi denotes the reduced Z2-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these [d/2] 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 C*(K)→C*(Rd).
CITATION STYLE
Goaoc, X., Paták, P., Patáková, Z., Tancer, M., & Wagner, U. (2017). Bounding helly numbers via betti numbers. In A Journey through Discrete Mathematics: A Tribute to Jiri Matousek (pp. 407–447). Springer International Publishing. https://doi.org/10.1007/978-3-319-44479-6_17
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