{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:01:44Z","timestamp":1787320904427,"version":"3.56.0"},"reference-count":21,"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":[[2016,1]]},"abstract":"<jats:p>A $k$-uniform family of subsets of $[n]$ is intersecting if it does not contain a disjoint pair of sets. The study of intersecting families is central to extremal set theory, dating back to the seminal Erd\u00f6s--Ko--Rado theorem of 1961 that bounds the size of the largest such families. A recent trend has been to investigate the structure of set families with few disjoint pairs. Friedgut and Regev proved a general removal lemma, showing that when $\\gamma n \\le k \\le (\\tfrac12 - \\gamma)n$, a set family with few disjoint pairs can be made intersecting by removing few sets. We provide a simple proof of a removal lemma for large families, showing that families of size close to $\\ell \\binom{n-1}{k-1}$ with relatively few disjoint pairs must be close to a union of $\\ell$ stars. Our lemma holds for a wide range of uniformities; in particular, when $\\ell = 1$, the result holds for all $2 \\le k &lt; \\frac{n}{2}$ and provides sharp quantitative estimates. We use this removal lemma to answer a question of Bollob\u00e1s, Narayanan, and Raigorodskii regarding the independence number of random subgraphs of the Kneser graph $K(n,k)$. The Erd\u00f6s--Ko--Rado theorem shows $\\alpha(K(n,k)) = \\binom{n-1}{k-1}$. For some constant $c &gt; 0$ and $k \\le cn$, we determine the sharp threshold for when this equality holds for random subgraphs of $K(n,k)$, and we provide strong bounds on the critical probability for $k \\le \\tfrac12 (n-3)$.<\/jats:p>","DOI":"10.1137\/15m105149x","type":"journal-article","created":{"date-parts":[[2016,5,25]],"date-time":"2016-05-25T13:52:24Z","timestamp":1464184344000},"page":"1102-1114","source":"Crossref","is-referenced-by-count":19,"title":["Removal and Stability for Erd\u00f6s--Ko--Rado"],"prefix":"10.1137","volume":"30","author":[{"given":"Shagnik","family":"Das","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tuan","family":"Tran","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2016,5,25]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548309990253"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1017\/fms.2015.21"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2015.01.003"},{"key":"atypb4","first-page":"91","author":"Baranyai Z.","year":"1975","journal-title":"Amsterdam"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2015.08.002"},{"key":"atypb6","first-page":"1","author":"Conlon D.","year":"2013","journal-title":"Cambridge"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"S. Das, W. Gan, and B. Sudakov,\n                      The minimum number of disjoint pairs in set systems and related problems\n                      , Combinatorica, to appear, http:\/\/dx.doi.org\/10.1007\/s00493-014-3133-0.","DOI":"10.1007\/s00493-014-3133-0"},{"key":"atypb8","first-page":"P1","volume":"22","author":"Das S.","year":"2015","journal-title":"Electron. J. Combin."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548308009309"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/12.1.313"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(87)90005-7"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-008-2318-9"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/S0196-8858(02)00024-6"},{"key":"atypb16","unstructured":"E. Friedgut and O. Regev,\n                      personal communication\n                      , 2014."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/18.1.369"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548311000587"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2008.03.023"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-010-2401-x"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1979.1055985"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548311000472"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548312000387"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/15M105149X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:05:31Z","timestamp":1787317531000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/15M105149X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1]]},"references-count":21,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2016,1]]}},"alternative-id":["10.1137\/15M105149X"],"URL":"https:\/\/doi.org\/10.1137\/15m105149x","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,1]]}}}