{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T16:49:34Z","timestamp":1773161374749,"version":"3.50.1"},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2024,12,11]],"date-time":"2024-12-11T00:00:00Z","timestamp":1733875200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2025,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline1.png\"\/><jats:tex-math>\n$T$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> be a tree on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline2.png\"\/><jats:tex-math>\n$t$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> vertices. We prove that for every positive integer <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline3.png\"\/><jats:tex-math>\n$k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and every graph <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline4.png\"\/><jats:tex-math>\n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, either <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline5.png\"\/><jats:tex-math>\n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> contains <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline6.png\"\/><jats:tex-math>\n$k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> pairwise vertex-disjoint subgraphs each having a <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline7.png\"\/><jats:tex-math>\n$T$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> minor, or there exists a set <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline8.png\"\/><jats:tex-math>\n$X$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> of at most <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline9.png\"\/><jats:tex-math>\n$t(k-1)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> vertices of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline10.png\"\/><jats:tex-math>\n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline11.png\"\/><jats:tex-math>\n$G-X$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> has no <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline12.png\"\/><jats:tex-math>\n$T$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> minor. The bound on the size of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline13.png\"\/><jats:tex-math>\n$X$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is best possible and improves on an earlier <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline14.png\"\/><jats:tex-math>\n$f(t)k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> bound proved by Fiorini, Joret, and Wood (2013) with some fast-growing function <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000415_inline15.png\"\/><jats:tex-math>\n$f(t)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Moreover, our proof is short and simple.<\/jats:p>","DOI":"10.1017\/s0963548324000415","type":"journal-article","created":{"date-parts":[[2024,12,11]],"date-time":"2024-12-11T03:07:49Z","timestamp":1733886469000},"page":"321-325","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["Tight bound for the Erd\u0151s\u2013P\u00f3sa property of tree minors"],"prefix":"10.1017","volume":"34","author":[{"given":"Vida","family":"Dujmovi\u0107","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gwena\u00ebl","family":"Joret","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Piotr","family":"Micek","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pat","family":"Morin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2024,12,11]]},"reference":[{"key":"S0963548324000415_ref3","doi-asserted-by":"publisher","DOI":"10.1145\/2488608.2488645"},{"key":"S0963548324000415_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548313000266"},{"key":"S0963548324000415_ref6","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1965-035-8"},{"key":"S0963548324000415_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(83)90079-5"},{"key":"S0963548324000415_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300001450"},{"key":"S0963548324000415_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(86)90030-4"},{"key":"S0963548324000415_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(91)90068-U"},{"key":"S0963548324000415_ref2","article-title":"A tight Erd\u0151s-P\u00f3sa function for planar minors","volume":"10","author":"Cames van Batenburg","year":"2019","journal-title":"Adv. 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Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}