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Bounding Helly Numbers via Betti Numbers
[chapter]
<span title="">2017</span>
<i title="Springer International Publishing">
A Journey Through Discrete Mathematics
</i>
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We show that very weak topological assumptions are enough to ensure the existence of a Hellytype 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 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 Z 2 -Betti numbers (with singular homology). These topological conditions are sharp: not
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... rolling 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 * (R d ). Both techniques are of independent interest.
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