{"paper":{"title":"Nowhere dense Ramsey sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Marcelo Sales, Vojt\\v{e}ch R\\\"odl","submitted_at":"2024-02-27T02:00:44Z","abstract_excerpt":"A set of points $S$ in Euclidean space $\\mathbb{R}^d$ is called \\textit{Ramsey} if any finite partition of $\\mathbb{R}^{\\infty}$ yields a monochromatic copy of $S$. While characterization of Ramsey set remains a major open problem in the area, a stronger ``density'' concept was considered in [J. Amer. Math. Soc. 3, 1--7, 1990]: If $S$ is a $d$-dimensional simplex, then for any $\\mu>0$ there is an integer $d:=d(S,\\mu)$ and finite configuration $X\\subseteq \\mathbb{R}^d$ such that any subconfiguration $Y\\subseteq X$ with $|Y|\\geq \\mu |X|$ contains a copy of $S$. Complementing this, here we show t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.17137","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/2402.17137/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"}