{"paper":{"title":"Connected-Intersecting Families of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aaron Berger, Harish Vemuri, Michael Doppelt, Pat Devlin, Ross Berkowitz, Sonali Durham, Tessa Murthy","submitted_at":"2019-01-06T22:35:12Z","abstract_excerpt":"For a graph property $\\mathcal{P}$ and a common vertex set $V = \\{1, 2, \\ldots, n\\}$, a family of graphs on $V$ is \\emph{$\\mathcal{P}$-intersecting} iff $G \\cap H$ satisfies $\\mathcal{P}$ for all $G,H$ in the family. Addressing a question of Chung, Graham, Frankl, and Shearer, we explore---for various $\\mathcal{P}$---the maximum cardinality among all $\\mathcal{P}$-intersecting families of graphs. In the connected-intersecting case, we resolve the question completely by a short linear algebraic proof showing this maximum is attained by taking all graphs containing a fixed spanning tree (though "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01616","kind":"arxiv","version":1},"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"}