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 $\mathcal{F}$ is a finite family of subsets of ${\mathbb{R}}^d$ such that ${\tilde{\beta }}_i\left(\bigcap \mathcal{G}\right)\le b$ for any $\mathcal{G}⊊\mathcal{F}$ and every $0\le i\le \lceil d/2\rceil -1$ then $\mathcal{F}$ has Helly number at most $h(b,d)$. Here ${\tilde{\beta }}_i$ denotes the reduced ${\mathbb{Z}}_2$-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these $\lceil d/2\rceil$ 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 $C_{\ast }(K)\to C_{\ast }({\mathbb{R}}^d)$.

29 pages, 8 figures