{"pk":64922,"title":"Unavoidable order-size pairs in hypergraphs -- positive forcing density","subtitle":null,"abstract":"Erdős, Füredi, Rothschild and Sós initiated a study of classes of graphs that forbid every induced subgraph on a given number \\(m\\) of vertices and number \\(f\\) of edges. Extending their notation to \\(r\\)-graphs, we write \\((n,e) \\to_r (m,f)\\) if every \\(r\\)-graph \\(G\\) on \\(n\\) vertices with \\(e\\) edges has an induced subgraph on \\(m\\) vertices and \\(f\\) edges. The forcing density of a pair \\((m,f)\\) is \\[ \\sigma_r(m,f) =\\left. \\limsup\\limits_{n \\to \\infty} \\frac{|\\{e : (n,e) \\to_r (m,f)\\}|}{\\binom{n}{r}} \\right. .\\] In the graph setting it is known that there are infinitely many pairs \\((m, f)\\) with positive forcing density. Weber asked if there is a pair of positive forcing density for \\(r\\geq 3\\) apart from the trivial ones \\((m, 0)\\) and \\((m, \\binom{m}{r})\\). Answering her question, we show that \\((6,10)\\) is such a pair for \\(r=3\\) and conjecture that it is the unique such pair. Further, we find necessary conditions for a pair to have positive forcing density, supporting this conjecture.\n \nMathematics Subject Classifications: 05C35, 05C65\n \nKeywords: Induced hypergraphs, forcing density","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":"Induced hypergraphs"},{"word":"forcing density"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/29v8h0bq","frozenauthors":[{"first_name":"Maria","middle_name":"","last_name":"Axenovich","name_suffix":"","institution":"Karlsruhe Institute of Technology, 76133 Karlsruhe, Germany","department":""},{"first_name":"József","middle_name":"","last_name":"Balogh","name_suffix":"","institution":"University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, U.S.A.","department":""},{"first_name":"Felix","middle_name":"Christian","last_name":"Clemen","name_suffix":"","institution":"Karlsruhe Institute of Technology, 76133 Karlsruhe, Germany","department":""},{"first_name":"Lea","middle_name":"","last_name":"Weber","name_suffix":"","institution":"Karlsruhe Institute of Technology, 76133 Karlsruhe, Germany","department":""}],"date_submitted":"2023-12-22T14:23:13Z","date_accepted":"2023-12-22T14:23:13Z","date_published":"2023-12-22T08:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64922/galley/49732/download/"}]}