@inbook{424,
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 Rd such that βi(∩G)≤b for any G⊊F and every 0 ≤ i ≤ [d/2]-1 then F has Helly number at most h(b, d). Here βi 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).},
author = {Goaoc, Xavier and Paták, Pavel and Patakova, Zuzana and Tancer, Martin and Wagner, Uli},
booktitle = {A Journey through Discrete Mathematics: A Tribute to Jiri Matousek},
editor = {Loebl, Martin and Nešetřil, Jaroslav and Thomas, Robin},
isbn = {978-331944479-6},
pages = {407 -- 447},
publisher = {Springer},
title = {{Bounding helly numbers via betti numbers}},
doi = {10.1007/978-3-319-44479-6_17},
year = {2017},
}