Bounding Helly numbers via Betti numbers [article]

Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, Uli Wagner
2016 arXiv   pre-print
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 R^d such that β̃_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 β̃_i denotes the reduced Z_2-Betti numbers (with singular homology). These topological conditions are sharp: not
more » ... ntrolling 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).
arXiv:1310.4613v3 fatcat:xdowys3xjbe7jjta3yxaeddjsq