{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T20:30:48Z","timestamp":1649104248814},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,10,9]],"date-time":"2014-10-09T00:00:00Z","timestamp":1412812800000},"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":[[2015,5]]},"abstract":"<jats:p>The purpose of this short problem paper is to raise the following extremal question on set systems: Which set systems of a given size maximise the number of (<jats:italic>n<\/jats:italic> + 1)-element chains in the power set <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548314000510_inline1\" \/><jats:tex-math>$\\mathcal{P}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>(1,2,.\u00a0.\u00a0.,<jats:italic>n<\/jats:italic>)? We will show that for each fixed \u03b1 &gt; 0 there is a family of \u03b12<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup> sets containing (\u03b1 + <jats:italic>o<\/jats:italic>(1))<jats:italic>n<\/jats:italic>! such chains, and that this is asymptotically best possible. For smaller set systems we conjecture that a \u2018tower of cubes\u2019 construction is extremal. We finish by mentioning briefly a connection to an extremal problem on posets and a variant of our question for the grid graph.<\/jats:p>","DOI":"10.1017\/s0963548314000510","type":"journal-article","created":{"date-parts":[[2014,10,10]],"date-time":"2014-10-10T13:59:06Z","timestamp":1412949546000},"page":"480-485","source":"Crossref","is-referenced-by-count":0,"title":["Set Systems Containing Many Maximal Chains"],"prefix":"10.1017","volume":"24","author":[{"given":"J. ROBERT","family":"JOHNSON","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"IMRE","family":"LEADER","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"PAUL A.","family":"RUSSELL","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,10,9]]},"reference":[{"key":"S0963548314000510_ref4","article-title":"Supersaturation in the Boolean lattice","volume":"14A","author":"Dove","year":"2014","journal-title":"Integers"},{"key":"S0963548314000510_ref2","volume-title":"Combinatorics","author":"Bollob\u00e1s","year":"1986"},{"key":"S0963548314000510_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02582924"},{"key":"S0963548314000510_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548311000587"},{"key":"S0963548314000510_ref3","doi-asserted-by":"crossref","unstructured":"Das S. , Gan W. and Sudakov B. Sperner's theorem and a problem of Erd\u0151s, Katona and Kleitman. Combin. Probab. Comput., to appear.","DOI":"10.1017\/S0963548314000273"},{"key":"S0963548314000510_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548311000472"},{"key":"S0963548314000510_ref6","first-page":"215","volume-title":"Theory of Graphs: Proc. Colloquium Held at Tihany, Hungary, September 1966","author":"Kleitman","year":"1968"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548314000510","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,21]],"date-time":"2019-04-21T00:47:18Z","timestamp":1555807638000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548314000510\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,10,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2015,5]]}},"alternative-id":["S0963548314000510"],"URL":"https:\/\/doi.org\/10.1017\/s0963548314000510","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,10,9]]}}}