{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T16:46:21Z","timestamp":1648917981756},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2008,9,12]],"date-time":"2008-09-12T00:00:00Z","timestamp":1221177600000},"content-version":"unspecified","delay-in-days":4944,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[1995,3]]},"abstract":"<jats:p>Ramsey numbers for matroids, which mimic properties of Ramsey numbers for graphs, have been denned as follows. Let <jats:italic>k<\/jats:italic> and <jats:italic>l<\/jats:italic> be positive integers. Then <jats:italic>n<\/jats:italic>(<jats:italic>k, l<\/jats:italic>) is the least positive integer <jats:italic>n<\/jats:italic> such that every connected matroid with <jats:italic>n<\/jats:italic> elements contains either a circuit with at least <jats:italic>k<\/jats:italic> elements or a cocircuit with at least <jats:italic>l<\/jats:italic> elements. We determine the largest known value of these numbers in the sense of maximizing both <jats:italic>k<\/jats:italic> and <jats:italic>l<\/jats:italic>. We also find extremal matroids with small circuits and cocircuits. Results on matroid connectivity, geometry, and extremal matroid theory are used here.<\/jats:p>","DOI":"10.1017\/s0963548300001486","type":"journal-article","created":{"date-parts":[[2008,9,12]],"date-time":"2008-09-12T11:15:47Z","timestamp":1221218147000},"page":"67-80","source":"Crossref","is-referenced-by-count":1,"title":["Some Small Circuit-Cocircuit Ramsey Numbers for Matroids"],"prefix":"10.1017","volume":"4","author":[{"given":"Fair Barbour","family":"Hurst","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Talmage James","family":"Reid","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2008,9,12]]},"reference":[{"key":"S0963548300001486_ref016","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1966-129-2"},{"key":"S0963548300001486_ref010","unstructured":"[10] Ne\u0161et\u0159il J. , Poljak S. and Turz\u00edk D. (1985) Special amalgams and Ramsey matroids. In: Lov\u00e1sz L. and Recski A. (eds.) Matroid Theory, Colloq. Math. Soc. J\u00e1nos Bolyai 40 267\u2013298."},{"key":"S0963548300001486_ref004","first-page":"463","article-title":"A combinatorial problem in geometry","volume":"2","author":"Erd\u0151s","year":"1935","journal-title":"Compositio Math."},{"key":"S0963548300001486_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/S0095-8956(81)80007-X"},{"key":"S0963548300001486_ref014","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(80)90075-1"},{"key":"S0963548300001486_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(77)90118-2"},{"key":"S0963548300001486_ref018","volume-title":"Matroid Theory","author":"Welsh","year":"1976"},{"key":"S0963548300001486_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1971-0288039-7"},{"key":"S0963548300001486_ref012","volume-title":"Matroid Theory","author":"Oxley","year":"1992"},{"key":"S0963548300001486_ref013","unstructured":"[13] Reid T. J. (submitted) Ramsey numbers for matroids."},{"key":"S0963548300001486_ref003","unstructured":"[3] Cunningham W. H. (1973) A combinatorial decomposition theory, Ph.D. thesis, University of Waterloo."},{"key":"S0963548300001486_ref005","doi-asserted-by":"publisher","DOI":"10.1137\/0129045"},{"key":"S0963548300001486_ref006","unstructured":"[6] Hurst F. and Reid T. J. (to appear) Ramsey numbers for cocircuits in matroids. Ars Combinatorica."},{"key":"S0963548300001486_ref017","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(87)90074-6"},{"key":"S0963548300001486_ref015","unstructured":"[15] Thomas R. (private communication)."},{"key":"S0963548300001486_ref002","volume-title":"Graphs and Digraphs","author":"Chartrand","year":"1986"},{"key":"S0963548300001486_ref011","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1981-003-9"},{"key":"S0963548300001486_ref008","first-page":"21","volume-title":"Contemporary Math.","volume":"147","author":"Kung","year":"1993"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548300001486","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,14]],"date-time":"2019-05-14T19:22:18Z","timestamp":1557861738000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548300001486\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,3]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1995,3]]}},"alternative-id":["S0963548300001486"],"URL":"https:\/\/doi.org\/10.1017\/s0963548300001486","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,3]]}}}