{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T03:11:24Z","timestamp":1772507484256,"version":"3.50.1"},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2012,12,21]],"date-time":"2012-12-21T00:00:00Z","timestamp":1356048000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2013,3]]},"abstract":"<jats:p>We prove that there is a constant<jats:italic>c<\/jats:italic>such that, for each positive integer<jats:italic>k<\/jats:italic>, every (2<jats:italic>k<\/jats:italic>+ 1) \u00d7 (2<jats:italic>k<\/jats:italic>+ 1) array<jats:italic>A<\/jats:italic>on the symbols (1,.\u00a0.\u00a0.,2<jats:italic>k<\/jats:italic>+1) with at most<jats:italic>c<\/jats:italic>(2<jats:italic>k<\/jats:italic>+1) symbols in every cell, and each symbol repeated at most<jats:italic>c<\/jats:italic>(2<jats:italic>k<\/jats:italic>+1) times in every row and column is<jats:italic>avoidable<\/jats:italic>; that is, there is a (2<jats:italic>k<\/jats:italic>+1) \u00d7 (2<jats:italic>k<\/jats:italic>+1) Latin square<jats:italic>S<\/jats:italic>on the symbols 1,.\u00a0.\u00a0.,2<jats:italic>k<\/jats:italic>+1 such that, for each<jats:italic>i,j<\/jats:italic>\u2208 {1,.\u00a0.\u00a0.,2<jats:italic>k<\/jats:italic>+1}, the symbol in position (<jats:italic>i,j<\/jats:italic>) of<jats:italic>S<\/jats:italic>does not appear in the corresponding cell in<jats:italic>A<\/jats:italic>. This settles the last open case of a conjecture by H\u00e4ggkvist. Using this result, we also show that there is a constant \u03c1, such that, for any positive integer<jats:italic>n<\/jats:italic>, if each cell in an<jats:italic>n<\/jats:italic>\u00d7<jats:italic>n<\/jats:italic>array<jats:italic>B<\/jats:italic>is assigned a set of<jats:italic>m<\/jats:italic>\u2264 \u03c1<jats:italic>n<\/jats:italic>symbols, where each set is chosen independently and uniformly at random from {1,.\u00a0.\u00a0.,<jats:italic>n<\/jats:italic>}, then the probability that<jats:italic>B<\/jats:italic>is avoidable tends to 1 as<jats:italic>n<\/jats:italic>\u2192 \u221e.<\/jats:p>","DOI":"10.1017\/s0963548312000570","type":"journal-article","created":{"date-parts":[[2012,12,21]],"date-time":"2012-12-21T10:25:51Z","timestamp":1356085551000},"page":"184-212","source":"Crossref","is-referenced-by-count":12,"title":["Avoiding Arrays of Odd Order by Latin Squares"],"prefix":"10.1017","volume":"22","author":[{"given":"LINA J.","family":"ANDR\u00c9N","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"CARL JOHAN","family":"CASSELGREN","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"LARS-DANIEL","family":"\u00d6HMAN","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2012,12,21]]},"reference":[{"key":"S0963548312000570_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(89)90092-7"},{"key":"S0963548312000570_ref9","doi-asserted-by":"crossref","first-page":"N21","DOI":"10.37236\/1022","article-title":"Regular spanning subgraphs of bipartite graphs of high minimum degree","volume":"14","author":"Csaba","year":"2007","journal-title":"Electron. J. Combin."},{"key":"S0963548312000570_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(96)00354-8"},{"key":"S0963548312000570_ref6","first-page":"257","article-title":"Avoidable partial Latin squares of order 4m+1","volume":"95","author":"Cavenagh","year":"2010","journal-title":"Ars Combin."},{"key":"S0963548312000570_ref4","first-page":"27","article-title":"Certain properties of nonnegative matrices and their permanents","volume":"211","author":"Br\u00e8gman","year":"1973","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"S0963548312000570_ref10","doi-asserted-by":"crossref","first-page":"R47","DOI":"10.37236\/1073","article-title":"Latin squares with forbidden entries","volume":"13","author":"Cutler","year":"2006","journal-title":"Electron. J. Combin."},{"key":"S0963548312000570_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2011.10.028"},{"key":"S0963548312000570_ref2","unstructured":"Andr\u00e9n L. J. (2010) On Latin squares and avoidable arrays. Doctoral thesis, Ume\u00e5 University."},{"key":"S0963548312000570_ref1","doi-asserted-by":"publisher","DOI":"10.1002\/0471722154"},{"key":"S0963548312000570_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s00026-011-0106-5"},{"key":"S0963548312000570_ref7","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0118(199708)25:4<257::AID-JGT3>3.0.CO;2-J"},{"key":"S0963548312000570_ref3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511984068"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548312000570","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,7,19]],"date-time":"2020-07-19T08:09:07Z","timestamp":1595146147000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548312000570\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,12,21]]},"references-count":12,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2013,3]]}},"alternative-id":["S0963548312000570"],"URL":"https:\/\/doi.org\/10.1017\/s0963548312000570","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,12,21]]}}}