{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,10,9]],"date-time":"2024-10-09T04:26:43Z","timestamp":1728448003451},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2024,3,5]],"date-time":"2024-03-05T00:00:00Z","timestamp":1709596800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2024,7]]},"abstract":"Abstract<\/jats:title>A result of Gy\u00e1rf\u00e1s [12] exactly determines the size of a largest monochromatic component in an arbitrary \n$r$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-colouring of the complete \n$k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-uniform hypergraph \n$K_n^k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> when \n$k\\geq 2$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and \n$k\\in \\{r-1,r\\}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We prove a result which says that if one replaces \n$K_n^k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in Gy\u00e1rf\u00e1s\u2019 theorem by any \u2018expansive\u2019 \n$k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-uniform hypergraph on \n$n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> vertices (that is, a \n$k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-uniform hypergraph \n$G$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> on \n$n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> vertices in which \n$e(V_1, \\ldots, V_k)\\gt 0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for all disjoint sets \n$V_1, \\ldots, V_k\\subseteq V(G)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with \n$|V_i|\\gt \\alpha$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for all \n$i\\in [k]$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on \n$r$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and \n$\\alpha$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms.<\/jats:p>Gy\u00e1rf\u00e1s\u2019 result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary \n$r$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-partite \n$r$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-uniform hypergraph \n$H$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with \n$n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> edges in which every set of \n$k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> edges has a common intersection. In this language, our result says that if one replaces the condition that every set of \n$k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> edges has a common intersection with the condition that for every collection of \n$k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> disjoint sets \n$E_1, \\ldots, E_k\\subseteq E(H)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with \n$|E_i|\\gt \\alpha$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, there exists \n$(e_1, \\ldots, e_k)\\in E_1\\times \\cdots \\times E_k$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that \n$e_1\\cap \\cdots \\cap e_k\\neq \\emptyset$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, then the smallest possible maximum degree of \n$H$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is essentially the same (within a small error term depending on \n$r$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and \n$\\alpha$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>). We prove our results in this dual setting.<\/jats:p>","DOI":"10.1017\/s096354832400004x","type":"journal-article","created":{"date-parts":[[2024,3,5]],"date-time":"2024-03-05T00:56:00Z","timestamp":1709600160000},"page":"467-483","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Large monochromatic components in expansive hypergraphs"],"prefix":"10.1017","volume":"33","author":[{"given":"Deepak","family":"Bal","sequence":"first","affiliation":[]},{"given":"Louis","family":"DeBiasio","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2024,3,5]]},"reference":[{"key":"S096354832400004X_ref15","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199807)12:4<381::AID-RSA5>3.0.CO;2-P"},{"key":"S096354832400004X_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2018.10.001"},{"key":"S096354832400004X_ref13","doi-asserted-by":"publisher","DOI":"10.1002\/jcd.21395"},{"key":"S096354832400004X_ref12","first-page":"62","article-title":"Partition coverings and blocking sets in hypergraphs","volume":"71","author":"Gy\u00e1rf\u00e1s","year":"1977","journal-title":"Commun. Comput. Autom. Inst. Hungarian Acad. Sci."},{"key":"S096354832400004X_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/S096354831400056X"},{"key":"S096354832400004X_ref10","doi-asserted-by":"publisher","DOI":"10.37236\/9824"},{"key":"S096354832400004X_ref8","doi-asserted-by":"publisher","DOI":"10.1137\/16M1069717"},{"key":"S096354832400004X_ref2","doi-asserted-by":"publisher","DOI":"10.37236\/6089"},{"key":"S096354832400004X_ref16","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548315000279"},{"key":"S096354832400004X_ref17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2020.105256"},{"key":"S096354832400004X_ref14","first-page":"3","article-title":"Large monochromatic components in edge colored graphs with a minimum degree condition","volume":"24","author":"Gy\u00e1rf\u00e1s","year":"2017","journal-title":"Electron. J. Comb."},{"key":"S096354832400004X_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-014-2935-4"},{"key":"S096354832400004X_ref3","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.23044"},{"key":"S096354832400004X_ref6","doi-asserted-by":"publisher","DOI":"10.1002\/jcd.21707"},{"key":"S096354832400004X_ref11","doi-asserted-by":"publisher","DOI":"10.37236\/9039"},{"key":"S096354832400004X_ref1","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.22491"},{"key":"S096354832400004X_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579271"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096354832400004X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T16:08:28Z","timestamp":1728403708000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096354832400004X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,5]]},"references-count":17,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,7]]}},"alternative-id":["S096354832400004X"],"URL":"https:\/\/doi.org\/10.1017\/s096354832400004x","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2024,3,5]]},"assertion":[{"value":"\u00a9 The Author(s), 2024. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}