{"paper":{"title":"Nerves, minors, and piercing numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andreas F. Holmsen, Minki Kim, Seunghun Lee","submitted_at":"2017-06-16T08:35:54Z","abstract_excerpt":"We make the first step towards a \"nerve theorem\" for graphs. Let $G$ be a simple graph and let $\\mathcal{F}$ be a family of induced subgraphs of $G$ such that the intersection of any members of $\\mathcal{F}$ is either empty or connected. We show that if the nerve complex of $\\mathcal{F}$ has non-vanishing homology in dimension three, then $G$ contains the complete graph on five vertices as a minor. As a consequence we confirm a conjecture of Goaoc concerning an extension of the planar $(p,q)$ theorem due to Alon and Kleitman: Let $\\mathcal{F}$ be a finite family of open connected sets in the p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.05181","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"}