{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T23:36:23Z","timestamp":1790206583946,"version":"4.1.0"},"reference-count":17,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"name":"National Key Research and Development Program of China","award":["2020YFA0712100"],"award-info":[{"award-number":["2020YFA0712100"]}]},{"name":"National Key Research and Development Program of China","award":["2018YFA0704703"],"award-info":[{"award-number":["2018YFA0704703"]}]},{"name":"Beijing Scholars Program"},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11801109"],"award-info":[{"award-number":["11801109"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971325"],"award-info":[{"award-number":["11971325"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>For an $r$-graph $G$, the minimum $(r-1)$-degree $\\delta(G)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(G)$ is contained in at least $t$ edges of $G$. Given an $r$-graph $F$, the codegree density $\\gamma(F)$ is the largest $\\gamma&gt;0$ such that there are $F$-free $r$-graphs $G$ on $n$ vertices with $\\delta(G)\\ge(\\gamma-o(1))n$. In this paper, we consider the codegree density of projective geometries. Employing the moment identity of a subset of $PG_{m}(q)$, we prove (1) $\\gamma(PG_{2}(q))=\\frac{1}{2}$ for prime power $q\\equiv2\\pmod{3}$; and (2) $\\gamma(PG_{3}(q))=\\frac{2}{3}$ for prime power $q\\equiv1\\pmod{2}$ or $q\\equiv2\\pmod{3}$. Our results partially solve an open problem proposed by Keevash and Zhao [ J. Combin. Theory Ser. B, 97 (2007), pp. 919--928]. Previously, the codegree density problems for projective geometries were settled only for $PG_{2}(2)$, $PG_{3}(2)$, $PG_{3}(3),$ and $PG_{2}(q)$ with odd prime power $q$.<\/jats:p>","DOI":"10.1137\/20m1385512","type":"journal-article","created":{"date-parts":[[2021,7,1]],"date-time":"2021-07-01T13:49:51Z","timestamp":1625147391000},"page":"1548-1556","source":"Crossref","is-referenced-by-count":2,"title":["On the Codegree Density of $PG_m(q)$"],"prefix":"10.1137","volume":"35","author":[{"given":"Tao","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gennian","family":"Ge","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,7,1]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2013.05.030"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1946-08715-7"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/130926997"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.endm.2017.06.067"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.37236\/1711"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548305006905"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2004.12.007"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20249"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"P. Keevash,\n                      Hypergraph Tur\u00e1n problems\n                      , in Surveys in Combinatorics 2011, London Math. Soc. Lecture Note Ser. 392, Cambridge University Press, Cambridge, 2011, pp. 83-139.","DOI":"10.1017\/CBO9781139004114.004"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-005-0042-2"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2007.01.004"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/18M1163956"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2005.06.001"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1006\/jcta.2002.3284"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2006.11.006"},{"key":"atypb16","first-page":"436","volume":"48","author":"Tur\u00e1n P.","year":"1941","journal-title":"Mat. Fiz Lapok"},{"key":"atypb17","unstructured":"T. Zhang and G. Ge,\n                      New Lower Bounds for the Tur\u00e1n Density of $PG_{m}(q)$\n                      , preprint,https:\/\/arxiv.org\/abs\/2006.15518, 2020."}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1385512","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:06:24Z","timestamp":1787321184000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1385512"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":17,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1385512"],"URL":"https:\/\/doi.org\/10.1137\/20m1385512","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}