{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T04:44:32Z","timestamp":1764132272870},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2002,10,9]],"date-time":"2002-10-09T00:00:00Z","timestamp":1034121600000},"content-version":"unspecified","delay-in-days":38,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2002,9]]},"abstract":"<jats:p>We consider <jats:italic>k<\/jats:italic>-uniform set systems over a universe of size <jats:italic>n<\/jats:italic> such that the size of each \npairwise intersection of sets lies in one of <jats:italic>s<\/jats:italic> residue classes mod <jats:italic>q<\/jats:italic>, but <jats:italic>k<\/jats:italic> does not lie in any \nof these <jats:italic>s<\/jats:italic> classes. A celebrated theorem of Frankl and Wilson [8] states that any such set \nsystem has size at most (<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup><jats:sub><jats:italic>s<\/jats:italic><\/jats:sub>) \nwhen <jats:italic>q<\/jats:italic> is prime. In a remarkable recent paper, Grolmusz [9] \nconstructed set systems of \nsuperpolynomial size \u03a9(exp(<jats:italic>c<\/jats:italic> log<jats:sup>2<\/jats:sup><jats:italic>n<\/jats:italic>\/log log <jats:italic>n<\/jats:italic>)) when <jats:italic>q<\/jats:italic> = 6. We \ngive a new, simpler construction achieving a slightly improved bound. Our construction \ncombines a technique of Frankl [6] of \u2018applying polynomials to set systems\u2019 with Grolmusz's \nidea of employing polynomials introduced by Barrington, Beigel and Rudich [5]. We also \nextend Frankl's original argument to arbitrary prime-power moduli: for any \u03b5 &gt; 0, we \nconstruct systems of size <jats:italic>n<\/jats:italic><jats:sup><jats:italic>s+g(s)<\/jats:italic><\/jats:sup>, where <jats:italic>g<\/jats:italic>(<jats:italic>s<\/jats:italic>) = \u03a9(<jats:italic>s<\/jats:italic><jats:sup>1\u2212\u03b5<\/jats:sup>). \nOur work overlaps with a very \nrecent technical report by Grolmusz [10].<\/jats:p>","DOI":"10.1017\/s0963548302005242","type":"journal-article","created":{"date-parts":[[2002,10,11]],"date-time":"2002-10-11T07:48:11Z","timestamp":1034322491000},"page":"475-486","source":"Crossref","is-referenced-by-count":5,"title":["Constructing Large Set Systems \nwith Given Intersection Sizes \nModulo Composite Numbers"],"prefix":"10.1017","volume":"11","author":[{"given":"SAMUEL","family":"KUTIN","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2002,10,9]]},"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548302005242","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,4]],"date-time":"2019-04-04T18:08:59Z","timestamp":1554401339000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548302005242\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,9]]},"references-count":0,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2002,9]]}},"alternative-id":["S0963548302005242"],"URL":"https:\/\/doi.org\/10.1017\/s0963548302005242","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,9]]}}}