{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T00:16:09Z","timestamp":1778717769321,"version":"3.51.4"},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1653,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2009,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Each relational structure <jats:italic>X<\/jats:italic> has an associated Gaifman graph, which endows <jats:italic>X<\/jats:italic> with the properties of a graph. If <jats:italic>x<\/jats:italic> is an element of <jats:italic>X<\/jats:italic>, let <jats:italic>B<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>x<\/jats:italic>) be the ball of radius <jats:italic>n<\/jats:italic> around <jats:italic>x<\/jats:italic>. Suppose that <jats:italic>X<\/jats:italic> is infinite, connected and of bounded degree. A first-order sentence <jats:italic>\u03d5<\/jats:italic> in the language of <jats:italic>X<\/jats:italic> is almost surely true (resp. a.s. false) for finite substructures of <jats:italic>X<\/jats:italic> if for every <jats:italic>x<\/jats:italic> \u2208 <jats:italic>X<\/jats:italic>, the fraction of substructures of <jats:italic>B<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>x<\/jats:italic>) satisfying <jats:italic>\u03d5<\/jats:italic> approaches 1 (resp. 0) as <jats:italic>n<\/jats:italic> approaches infinity. Suppose further that, for every finite substructure, <jats:italic>X<\/jats:italic> has a disjoint isomorphic substructure. Then every <jats:italic>\u03d5<\/jats:italic> is a.s. true or a.s. false for finite substructures of <jats:italic>X<\/jats:italic>. This is one form of the geometric zero-one law. We formulate it also in a form that does not mention the ambient infinite structure. In addition, we investigate various questions related to the geometric zero-one law.<\/jats:p>","DOI":"10.2178\/jsl\/1245158092","type":"journal-article","created":{"date-parts":[[2009,6,16]],"date-time":"2009-06-16T13:16:09Z","timestamp":1245158169000},"page":"929-938","source":"Crossref","is-referenced-by-count":2,"title":["A geometric zero-one law"],"prefix":"10.1017","volume":"74","author":[{"given":"Robert H.","family":"Gilman","sequence":"first","affiliation":[]},{"given":"Yuri","family":"Gurevich","sequence":"additional","affiliation":[]},{"given":"Alexei","family":"Miasnikov","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200003376_ref003","article-title":"Zero-one laws: Thesauri and parametric conditions","volume":"91","author":"Blass","year":"2007","journal-title":"Bulletin of European Association for Theoretical Computer Science"},{"key":"S0022481200003376_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(87)90019-3"},{"key":"S0022481200003376_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-2639-4_10"},{"key":"S0022481200003376_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71879-2"},{"key":"S0022481200003376_ref012","first-page":"637","article-title":"Kl+1-free graphs: asymptotic structure and a 0\u22121 law","volume":"303","author":"Kolaitis","year":"1987","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200003376_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-2080-7_27"},{"key":"S0022481200003376_ref011","first-page":"105","article-title":"A construction of vertex-transitive non-Cayley graphs","volume":"10","author":"Jajcay","year":"1994","journal-title":"Austalas. 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