{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T23:54:56Z","timestamp":1783036496429,"version":"3.54.6"},"reference-count":21,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":5397,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Random Struct Algorithms"],"published-print":{"date-parts":[[1992,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/nust001.gif\" xlink:title=\"chemical structure image\"\/> be a random <jats:italic>Q<jats:sup>n<\/jats:sup><\/jats:italic>\u201d\u2010process, that is let <jats:italic>Q<jats:sub>0<\/jats:sub><\/jats:italic> be the empty spanning subgraph of the cube <jats:italic>Q<jats:sup>n<\/jats:sup><\/jats:italic> and, for 1 \u2a7d <jats:italic>t<\/jats:italic> \u2a7d <jats:italic>M<\/jats:italic> = <jats:italic>nN<\/jats:italic>\/2 = <jats:italic>n<\/jats:italic>2<jats:sup><jats:italic>n<\/jats:italic>\u22121<\/jats:sup>, let the graph <jats:italic>Q<jats:sub>t<\/jats:sub><\/jats:italic> be obtained from <jats:italic>Q<\/jats:italic><jats:sub>t\u22121<\/jats:sub> by the random addition of an edge of <jats:italic>Q<\/jats:italic><jats:sub>n<\/jats:sub> not present in <jats:italic>Q<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic>\u22121<\/jats:sub>. When <jats:italic>t<\/jats:italic> is about <jats:italic>N<\/jats:italic>\/2, a typical <jats:italic>Q<jats:sub>t<\/jats:sub><\/jats:italic> undergoes a certain \u201cphase transition'': the component structure changes in a sudden and surprising way. Let <jats:italic>t<\/jats:italic> = (1 + \u03f5) <jats:italic>N<\/jats:italic>\/2 where \u03f5 is independent of <jats:italic>n<\/jats:italic>. Then all the components of a typical <jats:italic>Q<jats:sub>t<\/jats:sub><\/jats:italic> have <jats:italic>o(N)<\/jats:italic> vertices if \u03f5 &lt; 0, while if \u03f5 &gt; 0 then, as proved by Ajtai, Koml\u00f3s, and Szemer\u00e9di, a typical <jats:italic>Q<jats:sub>t<\/jats:sub><\/jats:italic> has a \u201cgiant\u201d component with at least \u03b1(\u03f5)<jats:italic>N<\/jats:italic> vertices, where \u03b1(\u03f5) &gt; 0. In this note we give essentially best possible results concerning the emergence of this giant component close to the time of phase transition. Our results imply that if \u03b7 &gt; 0 is fixed and <jats:italic>t<\/jats:italic> \u2a7d (1 \u2212 <jats:italic>n<\/jats:italic><jats:sup>\u2212\u03b7<\/jats:sup>) <jats:italic>N<\/jats:italic>\/2, then all components of a typical <jats:italic>Q<jats:sub>t<\/jats:sub><\/jats:italic> have at most <jats:italic>n<jats:sup>\u03b2(\u03b7)<\/jats:sup><\/jats:italic> vertices, where \u03b2(\u03b7) &gt; 0. More importantly, if 60(log <jats:italic>n<\/jats:italic>)<jats:sup>3<\/jats:sup>\/<jats:italic>n<\/jats:italic> \u2a7d \u03f5 = \u03f5<jats:italic>n<\/jats:italic> = <jats:italic>o<\/jats:italic>(1), then the largest component of a typical <jats:italic>Q<jats:sub>t<\/jats:sub><\/jats:italic> has about 2\u03f5<jats:italic>N<\/jats:italic> vertices, while the second largest component has order <jats:italic>O(n<\/jats:italic>\u03f5<jats:sup>\u22122<\/jats:sup>). Loosely put, the evolution of a typical <jats:italic>Q<jats:sup>n<\/jats:sup><\/jats:italic> process is such that shortly after time <jats:italic>N<\/jats:italic>\/2 the appearance of each new edge results in the giant component acquiring 4 new vertices.<\/jats:p>","DOI":"10.1002\/rsa.3240030106","type":"journal-article","created":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T18:17:04Z","timestamp":1180203424000},"page":"55-90","source":"Crossref","is-referenced-by-count":53,"title":["The Evolution of Random Subgraphs of the Cube"],"prefix":"10.1002","volume":"3","author":[{"given":"B.","family":"Bollob\u00e1s","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Y.","family":"Kohayakawa","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"T.","family":"\u0141uczak","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579276"},{"key":"e_1_2_1_3_2","first-page":"91","volume-title":"Combinatorial Mathematics","author":"Bollob\u00e1s B.","year":"1983"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.2307\/1999405"},{"key":"e_1_2_1_5_2","first-page":"xvi + 447","volume-title":"Random Graphs","author":"Bollob\u00e1s B.","year":"1985"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240010107"},{"key":"e_1_2_1_7_2","unstructured":"B.Bollob\u00e1s Y.Kohayakawa andT.\u0141uczak On the evolution of random Boolean functions to appear."},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(13)80135-7"},{"key":"e_1_2_1_9_2","unstructured":"B.Bollob\u00e1sandI.Leader Matchings in the cube to appear."},{"key":"e_1_2_1_10_2","first-page":"90","article-title":"On the probability of the connectedness of a random subgraph of the n\u2010cube (in Russian)","volume":"13","author":"Burtin Yu. 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