{"pk":64853,"title":"High-dimensional tennis balls","subtitle":null,"abstract":"We show that there exist constants $\\alpha,\\epsilon>0$ such that for every positive integer $n$ there is a continuous odd function $f:S^m\\to S^n$, with $m\\geq \\alpha n$, such that the $\\epsilon$-expansion of the image of $f$ does not contain a great circle. This result is motivated by a conjecture of Vitali Milman about well-complemented almost Euclidean subspaces of spaces uniformly isomorphic to $\\ell_2^n$.\n \nMathematics Subject Classifications: 46B09,  60C05\n \nKeywords: Anti-Ramsey, antipodal subsphere","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":"Anti-Ramsey"},{"word":"antipodal subsphere"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/8bs4n5hc","frozenauthors":[{"first_name":"Timothy","middle_name":"","last_name":"Gowers","name_suffix":"","institution":"Collège de France and the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge","department":""},{"first_name":"Katarzyna","middle_name":"","last_name":"Wyczesany","name_suffix":"","institution":"Department of Pure Mathematics, Tel Aviv University, Israel","department":""}],"date_submitted":"2022-06-25T20:24:24Z","date_accepted":"2022-06-25T20:24:24Z","date_published":"2022-06-30T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64853/galley/49663/download/"}]}