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 β~i(⋂G)≤b for any
G⊊F and every 0≤i≤⌈d/2⌉−1
then F has Helly number at most h(b,d). Here β~i
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).Comment: 29 pages, 8 figure