{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:43:18Z","timestamp":1740123798609,"version":"3.37.3"},"reference-count":12,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,8,7]],"date-time":"2023-08-07T00:00:00Z","timestamp":1691366400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,8,7]],"date-time":"2023-08-07T00:00:00Z","timestamp":1691366400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Order"],"published-print":{"date-parts":[[2024,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In extremal set theory our usual goal is to find the maximal size of a family of subsets of an <jats:italic>n<\/jats:italic>-element set satisfying a condition. A condition is called chain-dependent, if it is satisfied for a family if and only if it is satisfied for its intersections with the <jats:italic>n<\/jats:italic>! full chains. We introduce a method to handle problems with such conditions, then show how it can be used to prove three classic theorems. Then, a theorem about families containing no two sets such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$A\\subset B$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>A<\/mml:mi>\n                    <mml:mo>\u2282<\/mml:mo>\n                    <mml:mi>B<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\lambda \\cdot |A| \\le |B|$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03bb<\/mml:mi>\n                    <mml:mo>\u00b7<\/mml:mo>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:mi>A<\/mml:mi>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:mi>B<\/mml:mi>\n                    <mml:mo>|<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is proved. Finally, we investigate problems where instead of the size of the family, the number of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-chains is maximized. Our method is to define a weight function on the sets (or <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-chains) and use it in a double counting argument involving full chains.<\/jats:p>","DOI":"10.1007\/s11083-023-09644-8","type":"journal-article","created":{"date-parts":[[2023,8,7]],"date-time":"2023-08-07T07:02:13Z","timestamp":1691391733000},"page":"551-559","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Chain-dependent Conditions in Extremal Set Theory"],"prefix":"10.1007","volume":"41","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6154-7905","authenticated-orcid":false,"given":"D\u00e1niel T.","family":"Nagy","sequence":"first","affiliation":[]},{"given":"Kartal","family":"Nagy","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,8,7]]},"reference":[{"issue":"12","key":"9644_CR1","doi-asserted-by":"publisher","first-page":"898","DOI":"10.1090\/S0002-9904-1945-08454-7","volume":"52","author":"P Erd\u0151s","year":"1945","unstructured":"Erd\u0151s, P.: On a lemma of Littlewood and Offord. 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