{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:45:28Z","timestamp":1787323528795,"version":"build-2736575974"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["648509"],"award-info":[{"award-number":["648509"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2020,1]]},"abstract":"<jats:p>A seminal result of Hajnal and Szemer\u00e9di states that if a graph $G$ with $n$ vertices has minimum degree $\\delta(G) \\ge (r-1)n\/r$ for some integer $r \\ge 2$, then $G$ contains a $K_r$-factor, assuming $r$ divides $n$. Extremal examples which show optimality of the bound on $\\delta(G)$ are very structured and, in particular, contain large independent sets. In analogy to the Ramsey--Tur\u00e1n theory, Balogh, Molla, and Sharifzadeh initiated the study of how the absence of such large independent sets influences sufficient minimum degree. We show the following two related results: (a) For any $r &gt; \\ell \\ge 2$, if $G$ is a graph satisfying $\\delta(G) \\ge \\frac{r - \\ell}{r - \\ell + 1}n + \\Omega(n)$ and $\\alpha_\\ell(G) =o(n)$, that is, a largest $K_\\ell$-free induced subgraph has at most $o(n)$ vertices, then $G$ contains a $K_r$-factor. This is optimal for $\\ell = r - 1$ and extends a result of Balogh, Molla, and Sharifzadeh who considered the case $r = 3$. (b) If a graph $G$ satisfies $\\delta(G) = \\Omega(n)$ and $\\alpha_r^*(G) =o(n)$, that is, every induced $K_r$-free $r$-partite subgraph of $G$ has at least one vertex class of size $o(n)$, then it contains a $K_r$-factor. A similar statement is proven for a general graph $H$.<\/jats:p>","DOI":"10.1137\/18m1211970","type":"journal-article","created":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T12:50:35Z","timestamp":1585831835000},"page":"1001-1010","source":"Crossref","is-referenced-by-count":15,"title":["On a Ramsey--Tur\u00e1n Variant of the Hajnal--Szemer\u00e9di Theorem"],"prefix":"10.1137","volume":"34","author":[{"given":"Rajko","family":"Nenadov","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2982-2978","authenticated-orcid":true,"given":"Yanitsa","family":"Pehova","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,4,2]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1996.0020"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548318000196"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20670"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548318000366"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01895727"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"R. Diestel,\n                      Graph Theory\n                      , 5th ed., Springer, Berlin, 2017.","DOI":"10.1007\/978-3-662-53622-3"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1962-060-4"},{"key":"atypb8","first-page":"395","author":"Erd\u00f6s P.","year":"1970","journal-title":"Budapest"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(91)90007-7"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300001218"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579342"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.4064\/fm-25-1-379-387"},{"key":"atypb13","first-page":"601","author":"Hajnal A.","year":"1970","journal-title":"Budapest"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/080729657"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548317000268"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1007\/s004930070020"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(00)00279-X"},{"key":"atypb18","unstructured":"J. Koml\u00f3s and M. Simonovits,\n                      Szemer\u00e9di's regularity lemma and its applications in graph theory\n                      , in Combinatorics, Paul Erd\u00f6s is eighty, Vol. 2, Keszthely, 1993, Bolyai Soc. Math. Stud. 2, J\u00e1nos Bolyai Math. Soc., Budapest, 1996, pp. 295-352."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548397003106"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-009-2254-3"},{"key":"atypb21","unstructured":"M. Kwan,\n                      Almost all Steiner Triple Systems have Perfect Matchings\n                      , preprint, arXiv:1611.02246, 2016."},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/s00373-014-1410-8"},{"key":"atypb23","unstructured":"R. Montgomery,\n                      Embedding Bounded Degree Spanning Trees in Random Graphs\n                      , preprint, arXiv:1405.6559, 2014."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2008.10.002"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(92)90141-2"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(00)00214-4"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2012.04.006"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1211970","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:07:38Z","timestamp":1787321258000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1211970"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":27,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["10.1137\/18M1211970"],"URL":"https:\/\/doi.org\/10.1137\/18m1211970","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1]]}}}