Bounding Helly Numbers via Betti Numbers [chapter]

Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, Uli Wagner
<span title="">2017</span> <i title="Springer International Publishing"> A Journey Through Discrete Mathematics </i> &nbsp;
We show that very weak topological assumptions are enough to ensure the existence of a Hellytype 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 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 Z 2 -Betti numbers (with singular homology). These topological conditions are sharp: not
more &raquo; ... rolling 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 ). Both techniques are of independent interest.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-44479-6_17">doi:10.1007/978-3-319-44479-6_17</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7grpm6zbwrcvtbdpfkbwxawj7q">fatcat:7grpm6zbwrcvtbdpfkbwxawj7q</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180512153751/http://drops.dagstuhl.de/opus/volltexte/2015/5129/pdf/46.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/03/9b/039b6b98536ba1b226cee3daaaa2004cde3c196d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-44479-6_17"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>