Bounding Helly Numbers via Betti Numbers

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 $$\mathcal{F}$$ is a finite family of subsets of ℝ d $$\mathbb{R}^{d}$$ such that β ̃ i ⋂ 𝒢 ≤ b $$\tilde{\beta }_{i}\left (\bigcap \mathcal{G}\right ) \leq b$$ for any 𝒢 ⊊ ℱ $$\mathcal{G} \subsetneq \mathcal{F}$$ and every 0 ≤ i ≤ ⌈d∕2⌉− 1 then ℱ $$\mathcal{F}$$ has Helly number at most h(b, d). Here β ̃ i $$\tilde{\beta }_{i}$$ denotes the reduced ℤ 2 $$\mathbb{Z}_{2}$$ -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 ∗ ( ℝ d ) $$C_{{\ast}}(K) \rightarrow C_{{\ast}}(\mathbb{R}^{d})$$ .