{"pk":65021,"title":"Canonical theorems in geometric Ramsey theory","subtitle":null,"abstract":"In Euclidean Ramsey Theory usually we are looking for monochromatic configurations in the Euclidean space, whose points are colored with a fixed number of colors. In the canonical version, the number of colors is arbitrary, and we are looking for an `unavoidable' set of colorings of a finite configuration, that is, a set of colorings with the property that one of them always appears in any coloring of the space. This set definitely includes the monochromatic and the rainbow colorings. In the present paper, we prove the following two results of this type. First, for any acute triangle \\(T\\), and any coloring of \\(\\mathbb{R}^3\\), there is either a monochromatic or a rainbow copy of \\(T\\). Second, for every \\(m\\), there exists a sufficiently large \\(n\\) such that in any coloring of \\(\\mathbb{R}^n\\), there exists either a monochromatic or a rainbow \\(m\\)-dimensional unit hypercube. In the maximum norm, \\(\\ell_{\\infty}\\), we have a much stronger statement. For every finite \\(M\\), there exits an \\(n\\) such that in any coloring of \\(\\mathbb{R}_\\infty^n\\), there is either a monochromatic or a rainbow isometric copy of \\(M\\).\n \nMathematics Subject Classifications: 05D10, 05C55\n \nKeywords: Euclidean Ramsey theory, canonical Ramsey theorem, colorings of the space","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Euclidean Ramsey theory"},{"word":"canonical Ramsey theorem"},{"word":"colorings of the space"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/0d76t2c1","frozenauthors":[{"first_name":"Panna","middle_name":"","last_name":"Gehér","name_suffix":"","institution":"HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary -- Eötvös Loránd University, Budapest, Hungary","department":""},{"first_name":"Arsenii","middle_name":"","last_name":"Sagdeev","name_suffix":"","institution":"HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary -- KIT, Karlsruhe, Germany","department":""},{"first_name":"Géza","middle_name":"","last_name":"Tóth","name_suffix":"","institution":"HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary -- Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Budapest, Hungary","department":""}],"date_submitted":"2026-02-02T09:25:28Z","date_accepted":"2026-02-02T09:25:28Z","date_published":"2025-12-20T08:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/65021/galley/49831/download/"}]}