Abstrakt
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(intersection of G)<=b for any proper subset G of F and every 0 <= i <= ceil(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 Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach .