{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T09:28:16Z","timestamp":1771406896684,"version":"3.50.1"},"reference-count":22,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T00:00:00Z","timestamp":1765238400000},"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":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We show that for any integer\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline1.png\"\/>\n                        <jats:tex-math>$k\\ge 1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    there exists an integer\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline2.png\"\/>\n                        <jats:tex-math>$t_0(k)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    such that, for integers\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline3.png\"\/>\n                        <jats:tex-math>$t, k_1, \\ldots , k_{t+1}, n$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    with\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline4.png\"\/>\n                        <jats:tex-math>$t\\gt t_0(k)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline5.png\"\/>\n                        <jats:tex-math>$\\max \\{k_1, \\ldots , k_{t+1}\\}\\le k$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline6.png\"\/>\n                        <jats:tex-math>$n \\gt 2k(t+1)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , the following holds: If\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline7.png\"\/>\n                        <jats:tex-math>$F_i$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is a\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline8.png\"\/>\n                        <jats:tex-math>$k_i$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -uniform hypergraph with vertex set\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline9.png\"\/>\n                        <jats:tex-math>$[n]$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and more than\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline10.png\"\/>\n                        <jats:tex-math>$ \\binom{n}{k_i}-\\binom{n-t}{k_i} - \\binom{n-t-k}{k_i-1} + 1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    edges for all\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline11.png\"\/>\n                        <jats:tex-math>$i \\in [t+1]$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , then either\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline12.png\"\/>\n                        <jats:tex-math>$\\{F_1,\\ldots , F_{t+1}\\}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    admits a rainbow matching of size\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline13.png\"\/>\n                        <jats:tex-math>$t+1$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    or there exists\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline14.png\"\/>\n                        <jats:tex-math>$W\\in \\binom{[n]}{t}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    such that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline15.png\"\/>\n                        <jats:tex-math>$W$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    intersects\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline16.png\"\/>\n                        <jats:tex-math>$F_i$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for all\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline17.png\"\/>\n                        <jats:tex-math>$i\\in [t+1]$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . This may be viewed as a rainbow non-uniform extension of the classical Hilton-Milner theorem. We also show that the same holds for every\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline18.png\"\/>\n                        <jats:tex-math>$t$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S096354832510031X_inline19.png\"\/>\n                        <jats:tex-math>$n \\gt 2k^3t$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , generalizing a recent stability result of Frankl and Kupavskii on matchings to rainbow matchings.\n                  <\/jats:p>","DOI":"10.1017\/s096354832510031x","type":"journal-article","created":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T08:45:44Z","timestamp":1765269944000},"page":"255-268","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["On stability of rainbow matchings"],"prefix":"10.1017","volume":"35","author":[{"given":"Hongliang","family":"Lu","sequence":"first","affiliation":[{"name":"Xi\u2019an Jiaotong University"}]},{"given":"Yan","family":"Wang","sequence":"additional","affiliation":[{"name":"Shanghai Jiao Tong University"}]},{"given":"Xingxing","family":"Yu","sequence":"additional","affiliation":[{"name":"Georgia Institute of Technology"}]}],"member":"56","published-online":{"date-parts":[[2025,12,9]]},"reference":[{"key":"S096354832510031X_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2019.02.004"},{"key":"S096354832510031X_ref1","unstructured":"[1] Aharoni, R. and Howard, D. Size conditions for the existence of rainbow matching. http:\/\/math.colgate.edu\/~dmhoward\/rsc.pdf"},{"key":"S096354832510031X_ref4","doi-asserted-by":"publisher","DOI":"10.1002\/9780470277331"},{"key":"S096354832510031X_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2011.12.002"},{"key":"S096354832510031X_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(84)90193-6"},{"key":"S096354832510031X_ref16","doi-asserted-by":"publisher","DOI":"10.1090\/jams\/1027"},{"key":"S096354832510031X_ref19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2022.105700"},{"key":"S096354832510031X_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2014.01.003"},{"key":"S096354832510031X_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2012.02.004"},{"key":"S096354832510031X_ref5","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/27.1.25"},{"key":"S096354832510031X_ref15","doi-asserted-by":"publisher","DOI":"10.1017\/S096354831100068X"},{"key":"S096354832510031X_ref17","unstructured":"[17] Keevash, P. , Lifshitz, N. , Long, E. and Minzer, D. (2019) Sharp thresholds and expanded hypergraphs, arXiv: 2103.04604 [math.CO]."},{"key":"S096354832510031X_ref22","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(89)90074-5"},{"key":"S096354832510031X_ref7","doi-asserted-by":"publisher","DOI":"10.37236\/2176"},{"key":"S096354832510031X_ref14","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/18.1.369"},{"key":"S096354832510031X_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2022.08.002"},{"key":"S096354832510031X_ref18","doi-asserted-by":"publisher","DOI":"10.5070\/C63160414"},{"key":"S096354832510031X_ref21","doi-asserted-by":"publisher","DOI":"10.1007\/BF02582941"},{"key":"S096354832510031X_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2016.08.003"},{"key":"S096354832510031X_ref6","first-page":"93","article-title":"A problem on independent \n\n\n\n$r$\n\n\n-tuples","volume":"8","author":"Erd\u0151s","year":"1965","journal-title":"Ann. Univ. Sci. Budapest. E\u00f6tv\u00f6s Sect. Math."},{"key":"S096354832510031X_ref13","doi-asserted-by":"publisher","DOI":"10.19086\/da.14507"},{"key":"S096354832510031X_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2013.01.008"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354832510031X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T08:53:52Z","timestamp":1771404832000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354832510031X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,9]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S096354832510031X"],"URL":"https:\/\/doi.org\/10.1017\/s096354832510031x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,9]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}