{"pk":64898,"title":"A proof of Frankl's conjecture on cross-union families","subtitle":null,"abstract":"The families \\(\\mathcal{F}_0,\\ldots,\\mathcal{F}_s\\) of \\(k\\)-element subsets of \\([n]:=\\{1,2,\\ldots,n\\}\\) are called cross-union if there is no choice of \\(F_0\\in \\mathcal{F}_0, \\ldots, F_s\\in \\mathcal{F}_s\\) such that \\(F_0\\cup\\ldots\\cup F_s=[n]\\). A natural generalization of the celebrated Erdős-Ko-Rado theorem, due to Frankl and Tokushige, states that for \\(n\\le (s+1)k\\) the geometric mean of \\(\\lvert\\mathcal{F}_i\\rvert\\) is at most \\(\\binom{n-1}{k}\\). Frankl conjectured that the same should hold for the arithmetic mean under some mild conditions. We prove Frankl's conjecture in a strong form by showing that the unique (up to isomorphism) maximizer for the arithmetic mean of cross-union families is the natural one \\(\\mathcal{F}_0=\\ldots=\\mathcal{F}_s={[n-1]\\choose k}\\).\n \nMathematics Subject Classifications: 05D05\n \nKeywords: Extremal set theory, generalizations of Erdős-Ko-Rado, cross-union families, cross-intersecting families","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":"Extremal set theory"},{"word":"generalizations of Erdős-Ko-Rado"},{"word":"cross-union families"},{"word":"cross-intersecting families"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/6gt1p3j8","frozenauthors":[{"first_name":"Stijn","middle_name":"","last_name":"Cambie","name_suffix":"","institution":"Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS), Daejeon, South Korea","department":""},{"first_name":"Jaehoon","middle_name":"","last_name":"Kim","name_suffix":"","institution":"Department of Mathematical Sciences, KAIST, South Korea","department":""},{"first_name":"Hong","middle_name":"","last_name":"Liu","name_suffix":"","institution":"Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS), Daejeon, South Korea","department":""},{"first_name":"Tuan","middle_name":"","last_name":"Tran","name_suffix":"","institution":"School of Mathematical Sciences, University of Science and Technology of China, China","department":""}],"date_submitted":"2023-09-14T07:37:37Z","date_accepted":"2023-09-14T07:37:37Z","date_published":"2023-09-15T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64898/galley/49708/download/"}]}