{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T13:57:41Z","timestamp":1788271061002,"version":"build-2803163510"},"reference-count":5,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2014,1]]},"abstract":"<jats:p>Given a set of $n$ real numbers, if the sum of the elements of every subset of size larger than $k$ is negative, what is the maximum number of subsets of nonnegative sum? In this note we show that the answer is $\\binom{n-1}{k-1} + \\binom{n-1}{k-2} + \\cdots + \\binom{n-1}{0}+1$, settling a problem of Tsukerman. We provide two proofs; the first establishes and applies a weighted version of Hall's theorem, and the second is based on an extension of the nonuniform Erd\u00f6s--Ko--Rado theorem.<\/jats:p>","DOI":"10.1137\/130947295","type":"journal-article","created":{"date-parts":[[2014,5,29]],"date-time":"2014-05-29T11:54:22Z","timestamp":1401364462000},"page":"811-816","source":"Crossref","is-referenced-by-count":3,"title":["Maximizing the Number of Nonnegative Subsets"],"prefix":"10.1137","volume":"28","author":[{"given":"Noga","family":"Alon","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Harout","family":"Aydinian","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hao","family":"Huang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2014,5,29]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2011.12.002"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/12.1.313"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-10.37.26"},{"key":"atypb4","first-page":"385","author":"Manickam N.","year":"1988","journal-title":"Amsterdam"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(88)90077-5"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/130947295","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:12:18Z","timestamp":1787317938000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/130947295"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,1]]},"references-count":5,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2014,1]]}},"alternative-id":["10.1137\/130947295"],"URL":"https:\/\/doi.org\/10.1137\/130947295","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,1]]}}}