{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T14:25:53Z","timestamp":1775053553570,"version":"3.50.1"},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2016,3,3]],"date-time":"2016-03-03T00:00:00Z","timestamp":1456963200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2016,9]]},"abstract":"<jats:p>Let <jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub> \u00d7 <jats:italic>G<\/jats:italic><jats:sub>2<\/jats:sub> denote the strong product of graphs <jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>G<\/jats:italic><jats:sub>2<\/jats:sub>, that is, the graph on <jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic><jats:sub>1<\/jats:sub>) \u00d7 <jats:italic>V<\/jats:italic>(<jats:italic>G<\/jats:italic><jats:sub>2<\/jats:sub>) in which (<jats:italic>u<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>u<\/jats:italic><jats:sub>2<\/jats:sub>) and (<jats:italic>v<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>v<\/jats:italic><jats:sub>2<\/jats:sub>) are adjacent if for each <jats:italic>i<\/jats:italic> = 1, 2 we have <jats:italic>u<jats:sub>i<\/jats:sub><\/jats:italic> = <jats:italic>v<jats:sub>i<\/jats:sub><\/jats:italic> or <jats:italic>u<\/jats:italic><jats:sub>i<\/jats:sub><jats:italic>v<\/jats:italic><jats:sub>i<\/jats:sub> \u2208 <jats:italic>E<\/jats:italic>(<jats:italic>G<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>). The Shannon capacity of <jats:italic>G<\/jats:italic> is <jats:italic>c<\/jats:italic>(<jats:italic>G<\/jats:italic>) = lim<jats:sub>n \u2192 \u221e<\/jats:sub> \u03b1(<jats:italic>G<jats:sup>n<\/jats:sup><\/jats:italic>)<jats:sup>1\/<jats:italic>n<\/jats:italic><\/jats:sup>, where <jats:italic>G<jats:sup>n<\/jats:sup><\/jats:italic> denotes the <jats:italic>n<\/jats:italic>-fold strong power of <jats:italic>G<\/jats:italic>, and \u03b1(<jats:italic>H<\/jats:italic>) denotes the independence number of a graph <jats:italic>H<\/jats:italic>. The normalized Shannon capacity of <jats:italic>G<\/jats:italic> is\n<jats:disp-formula-group><jats:disp-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548316000055_eqnU1\"\/><jats:tex-math>$$C(G) = \\ffrac {\\log c(G)}{\\log |V(G)|}.$$<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula><\/jats:disp-formula-group>\nAlon [1] asked whether for every \u03b5 &lt; 0 there are graphs <jats:italic>G<\/jats:italic> and <jats:italic>G<\/jats:italic>\u2032 satisfying <jats:italic>C<\/jats:italic>(<jats:italic>G<\/jats:italic>), <jats:italic>C<\/jats:italic>(<jats:italic>G<\/jats:italic>\u2032) &lt; \u03b5 but with <jats:italic>C<\/jats:italic>(<jats:italic>G<\/jats:italic> + <jats:italic>G<\/jats:italic>\u2032) &gt; 1 \u2212 \u03b5. We show that the answer is no.<\/jats:p>","DOI":"10.1017\/s0963548316000055","type":"journal-article","created":{"date-parts":[[2016,3,3]],"date-time":"2016-03-03T05:12:00Z","timestamp":1456981920000},"page":"766-767","source":"Crossref","is-referenced-by-count":8,"title":["On the Normalized Shannon Capacity of a Union"],"prefix":"10.1017","volume":"25","author":[{"given":"PETER","family":"KEEVASH","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"EOIN","family":"LONG","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,3,3]]},"reference":[{"key":"S0963548316000055_ref6","doi-asserted-by":"publisher","DOI":"10.1109\/18.720537"},{"key":"S0963548316000055_ref2","unstructured":"Alon N. (2002) Graph powers, Contemporary Combinatorics, Vol. 10 of Bolyai Society Mathematical Studies, J\u00e1nos Bolyai Mathematical Society, pp. 11\u201328."},{"key":"S0963548316000055_ref4","doi-asserted-by":"publisher","DOI":"10.1109\/18.412676"},{"key":"S0963548316000055_ref3","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2006.872856"},{"key":"S0963548316000055_ref7","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1979.1055985"},{"key":"S0963548316000055_ref5","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1979.1056027"},{"key":"S0963548316000055_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/PL00009824"},{"key":"S0963548316000055_ref8","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1956.1056798"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548316000055","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,18]],"date-time":"2019-04-18T20:04:20Z","timestamp":1555617860000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548316000055\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,3,3]]},"references-count":8,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2016,9]]}},"alternative-id":["S0963548316000055"],"URL":"https:\/\/doi.org\/10.1017\/s0963548316000055","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,3,3]]}}}