{"paper":{"title":"Intersection patterns in spaces with a forbidden homological minor","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"Andreas F. Holmsen, Xavier Goaoc, Zuzana Pat\\'akov\\'a","submitted_at":"2021-03-16T19:05:54Z","abstract_excerpt":"In this paper we study generalizations of classical results on intersection patterns of set systems in $\\mathbb{R}^d$, such as the fractional Helly theorem or the $(p,q)$-theorem, in the setting of arbitrary triangulable spaces with a forbidden homological minor.\n  Given a simplicial complex $K$ and an integer $b$, we say that a family $\\mathcal{F}$ of subcomplexes of some simplicial complex $X$ is a $(K,b)$-free cover if (i) $K$ is a forbidden homological minor of $X$, and (ii) the $j$th reduced Betti number $\\tilde{\\beta}_j(\\bigcap_{S\\in {\\mathcal{G}}}S,\\mathbb{Z}_2)$ is strictly less than $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2103.09286","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2103.09286/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}