Bounding Helly Numbers via Betti Numbers

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 such that β̃i ( ⋂G) ≤ b for any G ( F and every 0 ≤ i ≤ dd/2e−1 then F has Helly number at most h(b, d). Here β̃i denotes the reduced Z2-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these dd/2e 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).