{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T06:39:32Z","timestamp":1759991972587},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2015,1,20]],"date-time":"2015-01-20T00:00:00Z","timestamp":1421712000000},"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":[[2015,9]]},"abstract":"<jats:p>The<jats:italic>shadow<\/jats:italic>of a system of sets is all sets which can be obtained by taking a set in the original system, and removing a single element. The Kruskal-Katona theorem tells us the minimum possible size of the shadow of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548314000790_inline1\" \/><jats:tex-math>$\\mathcal A$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, if<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548314000790_inline1\" \/><jats:tex-math>$\\mathcal A$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>consists of<jats:italic>m r<\/jats:italic>-element sets.<\/jats:p><jats:p>In this paper, we ask questions and make conjectures about the minimum possible size of a<jats:italic>partial shadow<\/jats:italic>for<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548314000790_inline1\" \/><jats:tex-math>$\\mathcal A$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, which contains most sets in the shadow of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548314000790_inline1\" \/><jats:tex-math>$\\mathcal A$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. For example, if<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548314000790_inline2\" \/><jats:tex-math>$\\mathcal B$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is a family of sets containing all but one set in the shadow of each set of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548314000790_inline1\" \/><jats:tex-math>$\\mathcal A$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, how large must<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0963548314000790_inline2\" \/><jats:tex-math>$\\mathcal B$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>be?<\/jats:p>","DOI":"10.1017\/s0963548314000790","type":"journal-article","created":{"date-parts":[[2015,1,20]],"date-time":"2015-01-20T11:27:36Z","timestamp":1421753256000},"page":"825-828","source":"Crossref","is-referenced-by-count":2,"title":["Partial Shadows of Set Systems"],"prefix":"10.1017","volume":"24","author":[{"given":"B\u00c9LA","family":"BOLLOB\u00c1S","sequence":"first","affiliation":[]},{"given":"TOM","family":"ECCLES","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2015,1,20]]},"reference":[{"key":"S0963548314000790_ref2","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1525\/9780520319875-014","volume-title":"Mathematical Optimization Techniques","author":"Kruskal","year":"1963"},{"key":"S0963548314000790_ref3","volume-title":"Combinatorial Problems and Exercises","author":"Lov\u00e1sz","year":"1979"},{"key":"S0963548314000790_ref1","first-page":"187","volume-title":"Theory of Graphs","author":"Katona","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\/S0963548314000790","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T02:30:28Z","timestamp":1651026628000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548314000790\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1,20]]},"references-count":3,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2015,9]]}},"alternative-id":["S0963548314000790"],"URL":"https:\/\/doi.org\/10.1017\/s0963548314000790","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1,20]]}}}