{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T04:27:26Z","timestamp":1778732846521,"version":"3.51.4"},"reference-count":11,"publisher":"Wiley","issue":"1-2","license":[{"start":{"date-parts":[[2017,3,31]],"date-time":"2017-03-31T00:00:00Z","timestamp":1490918400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2017,4]]},"abstract":"<jats:p>We study finite \u2113\u2010colourable structures with an underlying pregeometry. The probability measure that is used corresponds to a process of generating such structures by which colours are first randomly assigned to all 1\u2010dimensional subspaces and then relationships are assigned in such a way that the colouring conditions are satisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure, where we now forget the specific colouring of the generating process, has a given property. With this measure we get the following results: (1) A zero\u2010one law. (2) The set of sentences with asymptotic probability 1 has an explicit axiomatisation which is presented. (3) There is a formula <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201500006-math-0001.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201500006:malq201500006-math-0001\"\/> (not directly speaking about colours) such that, with asymptotic probability 1, the relation \u201cthere is an \u2113\u2010colouring which assigns the same colour to <jats:italic>x<\/jats:italic> and <jats:italic>y<\/jats:italic>\u201d is defined by <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201500006-math-0002.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201500006:malq201500006-math-0002\"\/>. (4) With asymptotic probability 1, an \u2113\u2010colourable structure has a unique \u2113\u2010colouring (up to permutation of the colours).<\/jats:p>","DOI":"10.1002\/malq.201500006","type":"journal-article","created":{"date-parts":[[2017,3,31]],"date-time":"2017-03-31T15:57:00Z","timestamp":1490975820000},"page":"32-58","source":"Crossref","is-referenced-by-count":2,"title":["Random \u2113\u2010colourable structures with a pregeometry"],"prefix":"10.1002","volume":"63","author":[{"given":"Ove","family":"Ahlman","sequence":"first","affiliation":[{"name":"Department of Mathematics Uppsala University Postal Box 480 Uppsala 75106 Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vera","family":"Koponen","sequence":"additional","affiliation":[{"name":"Department of Mathematics Uppsala University Postal Box 480 Uppsala 75106 Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2017,3,31]]},"reference":[{"key":"e_1_2_8_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-012-2657-4"},{"key":"e_1_2_8_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2005.05.013"},{"key":"e_1_2_8_4_1","volume-title":"Perspectives in Mathematical Logic","author":"Ebbinghaus H.\u2010D.","year":"1999"},{"key":"e_1_2_8_5_1","unstructured":"P.Erd\u0151s D. J.Kleitman andB. L.Rothschild Asymptotic enumeration of\u2010free graphs in:Proceedings of Colloquio Internazionale sulle Teorie Combinatorie held in Rome 1973(Accademia Nazionale dei Lincei 1976) pp.19\u201327."},{"key":"e_1_2_8_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(72)90005-9"},{"key":"e_1_2_8_7_1","volume-title":"Wiley\u2010Interscience Series in Discrete Mathematics","author":"Graham R. 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