{"paper":{"title":"Transforming Rectangles into Squares, with Applications to Strong Colorings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Assaf Rinot","submitted_at":"2011-03-15T03:44:30Z","abstract_excerpt":"It is proved that every singular cardinal $\\lambda$ admits a function $RTS:[\\lambda^+]^2\\rightarrow[\\lambda^+]^2$ that transforms rectangles into squares. Namely, for every cofinal subsets $A,B$ of $\\lambda^+$, there exists a cofinal subset $C$ of $lambda^+$, such that $RTS[AxB]$ covers CxC.\n  When combined with a recent result of Eisworth, this shows that Shelah's notion of strong coloring $Pr_1(\\lambda^+,\\lambda^+,\\lambda^+,\\cf(\\lambda))$ coincides with the classical negative partition relation $\\lambda^+\\not\\rightarrow[\\lambda^+]^2_{\\lambda^+}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.2838","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"}