@inproceedings{1512,
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 F is a finite family of subsets of R^d such that the ith reduced Betti number (with Z_2 coefficients in singular homology) of the intersection of any proper subfamily G of F is at most b for every non-negative integer i less or equal to (d-1)/2, then F has Helly number at most h(b,d). These topological conditions are sharp: not controlling any of these 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 from C_*(K) to C_*(R^d). Both techniques are of independent interest.},
author = {Goaoc, Xavier and Paták, Pavel and Patakova, Zuzana and Tancer, Martin and Wagner, Uli},
location = {Eindhoven, Netherlands},
pages = {507 -- 521},
publisher = {ACM},
title = {{Bounding Helly numbers via Betti numbers}},
doi = {10.4230/LIPIcs.SOCG.2015.507},
volume = {34},
year = {2015},
}