{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T07:42:46Z","timestamp":1694763766750},"reference-count":11,"publisher":"Wiley","issue":"6","license":[{"start":{"date-parts":[[2018,12,17]],"date-time":"2018-12-17T00:00:00Z","timestamp":1545004800000},"content-version":"vor","delay-in-days":16,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2018,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Given a countable graph, we say a set <jats:italic>A<\/jats:italic> of its vertices is <jats:italic>universal<\/jats:italic> if it contains every countable graph as an induced subgraph, and <jats:italic>A<\/jats:italic> is <jats:italic>weakly universal<\/jats:italic> if it contains every finite graph as an induced subgraph. We show that, for almost every graph on <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700008-math-0001.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700008:malq201700008-math-0001\" \/>, (1) every set of positive upper density is universal, and (2) every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700008-math-0002.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700008:malq201700008-math-0002\" \/> will almost surely contain non\u2010universal sets with divergent reciprocal sums) and, more generally, that neither of these two results holds for a large class of partition regular families.<\/jats:p>","DOI":"10.1002\/malq.201700008","type":"journal-article","created":{"date-parts":[[2018,12,17]],"date-time":"2018-12-17T15:09:11Z","timestamp":1545059351000},"page":"478-486","update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Which subsets of an infinite random graph look random?"],"prefix":"10.1002","volume":"64","author":[{"given":"Will","family":"Brian","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics University of North Carolina at Charlotte 9201 University City Blvd. Charlotte NC 28223\u20100001 United States of America"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2018,12,17]]},"reference":[{"key":"e_1_2_4_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2668-8"},{"key":"e_1_2_4_3_1","doi-asserted-by":"crossref","first-page":"681","DOI":"10.2307\/44153760","article-title":"On the structure of measurable filters on a countable set","volume":"17","author":"Bartoszynski T.","year":"1992","journal-title":"Real Anal. 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