{"paper":{"title":"Bounding Helly numbers via Betti numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","cs.DM","math.AT"],"primary_cat":"math.CO","authors_text":"Martin Tancer, Pavel Pat\\'ak, Uli Wagner, Xavier Goaoc, Zuzana Pat\\'akov\\'a","submitted_at":"2013-10-17T08:23:28Z","abstract_excerpt":"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 $\\mathcal F$ is a finite family of subsets of $\\mathbb R^d$ such that $\\tilde\\beta_i\\left(\\bigcap\\mathcal G\\right) \\le b$ for any $\\mathcal G \\subsetneq \\mathcal F$ and every $0 \\le i \\le \\lceil d/2 \\rceil-1$ then $\\mathcal F$ has Helly number at most $h(b,d)$. Here $\\tilde\\beta_i$ denotes the reduced $\\mathbb Z_2$-Betti numbers (with singular homology)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.4613","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}