{"paper":{"title":"On a colorful problem by Dol'nikov concerning translates of convex bodies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Edgardo Rold\\'an-Pensado, Leonardo Mart\\'inez-Sandoval","submitted_at":"2023-07-15T05:21:29Z","abstract_excerpt":"In this note we study a conjecture by Jer\\'onimo-Castro, Magazinov and Sober\\'on which generalized a question posed by Dol'nikov. Let $F_1,F_2,\\dots,F_n$ be families of translates of a convex compact set $K$ in the plane so that each two sets from distinct families intersect. We show that, for some $j$, $\\bigcup_{i\\neq j}F_i$ can be pierced by at most $4$ points. To do so, we use previous ideas from Gomez-Navarro and Rold\\'an-Pensado together with an approximation result closely tied to the Banach-Mazur distance to the square."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.07714","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/2307.07714/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"}