{"paper":{"title":"Transversals to colorful intersecting convex sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cuauhtemoc Gomez-Navarro, Edgardo Rold\\'an-Pensado","submitted_at":"2023-05-26T09:22:32Z","abstract_excerpt":"Let $K$ be a compact convex set in $\\mathbb{R}^2$ and let $\\mathcal{F}_1, \\mathcal{F}_2, \\mathcal{F}_3$ be finite families of translates of $K$ such that $A \\cap B \\neq \\emptyset$ for every $A \\in \\mathcal{F}_i$ and $B \\in \\mathcal{F}_j$ with $i \\neq j$. A conjecture by Dolnikov is that, under these conditions, there is always some $j \\in \\lbrace 1,2,3 \\rbrace$ such that $\\mathcal{F}_j$ can be pierced by $3$ points. In this paper we prove a stronger version of this conjecture when $K$ is a body of constant width or when it is close in Banach-Mazur distance to a disk. We also show that the conj"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.16760","kind":"arxiv","version":2},"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/2305.16760/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"}