{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,26]],"date-time":"2026-09-26T12:02:38Z","timestamp":1790424158729,"version":"4.1.0"},"reference-count":49,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1600390"],"award-info":[{"award-number":["DMS-1600390"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1855653"],"award-info":[{"award-number":["DMS-1855653"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1600483"],"award-info":[{"award-number":["DMS-1600483"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1855622"],"award-info":[{"award-number":["DMS-1855622"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CNS 1229081"],"award-info":[{"award-number":["CNS 1229081"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1205413"],"award-info":[{"award-number":["1205413"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["AGS 0835579"],"award-info":[{"award-number":["AGS 0835579"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CNS 0958354"],"award-info":[{"award-number":["CNS 0958354"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CNS-0821794"],"award-info":[{"award-number":["CNS-0821794"]}],"id":[{"id":"10.13039\/100000001","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>Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method was developed to find asymptotic results for very large graphs, so it seems that the method is not suitable for finding small Ramsey numbers. But this intuition is wrong, and we will develop a technique to do just that in this paper. We find new upper bounds for many small graph and hypergraph Ramsey numbers. As a result, we prove the exact values $R(K_4^-,K_4^-,K_4^-)=28$, $R(K_8,C_5)= 29$, $R(K_9,C_6)= 41$, $R(Q_3,Q_3)=13$, $R(K_{3,5},K_{1,6})=17$, $R(C_3, C_5, C_5)= 17$, and $R(K_4^-,K_5^-;3)= 12$. We hope that this technique will be adapted to address other questions for smaller graphs with the flag algebra method.<\/jats:p>","DOI":"10.1137\/18m1169473","type":"journal-article","created":{"date-parts":[[2021,10,6]],"date-time":"2021-10-06T14:59:19Z","timestamp":1633532359000},"page":"2328-2344","source":"Crossref","is-referenced-by-count":16,"title":["Semidefinite Programming and Ramsey Numbers"],"prefix":"10.1137","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8612-3594","authenticated-orcid":true,"given":"Bernard","family":"Lidick\u00fd","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0124-3421","authenticated-orcid":true,"given":"Florian","family":"Pfender","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,10,6]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/15M1041754"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548310000222"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2015.08.006"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548314000820"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1080\/10556789908805765"},{"key":"atypb6","unstructured":"L. Boza,\n                      personal communication\n                      , 2012."},{"key":"atypb7","unstructured":"L. Boza,\n                      Sobre el. n\u00famero de Ramsey $R(K_6-e , K_4)$\n                      , in VIII Encuentro Andaluz de Matem\u00e1tica Discreta, Sevilla, Spain, 2013."},{"key":"atypb8","unstructured":"L. Boza,\n                      Sobre los n\u00fameros de Ramsey $R(K_5-e , K_5)$ y $R(K_6-e , K_4)$\n                      , in IX Jornada de Matem\u00e1tica Discreta y Algoritmica, JMDA 2014, Tarragona, Spain, 2014."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.37236\/2160"},{"key":"atypb10","unstructured":"L. Boza and J. Portillo,\n                      Sobre el n\u00famero de Ramsey $R(K_4-e ,K_7)$\n                      , in Proceedings of the VIII Jornada de Matem\u00e1tica Discreta y Algoritmica, 2012."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190130104"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2013.05.002"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2013.02.003"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.37236\/641"},{"key":"atypb15","first-page":"125","volume":"9","author":"Erd\u00f6s P.","year":"1964","journal-title":"Magyar Tud. Akad. Mat. Kutat\u00f3 Int. K\u00f6zl."},{"key":"atypb16","first-page":"5","volume":"2","author":"Exoo G.","year":"1987","journal-title":"J. Combin. Math. Combin. Comput."},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(91)90122-I"},{"key":"atypb18","unstructured":"G. Exoo,\n                      New lower bounds for table iii\n                      , Technical report, Indiana State University, Terre Haute, IN, 2000; constructions available athttp:\/\/cs.indstate.edu\/ge\/RAMSEY."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/130926997"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.22041"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2014.07.007"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-015-1267-4"},{"key":"atypb23","first-page":"300","author":"Goaoc X.","year":"2015","journal-title":"Germany"},{"key":"atypb24","first-page":"197","volume":"70","author":"Harborth H.","year":"2006","journal-title":"Util. Math."},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2012.12.008"},{"key":"atypb26","unstructured":"Y. Huang, F. Tan, J. Huang, and K. Zhang,\n                      New Upper Bounds for Ramsey Number $R(K_m-e , K_n-e)$\n                      , manuscript, 2016."},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1006\/eujc.1997.0186"},{"key":"atypb28","first-page":"42","volume":"1","author":"Jaradat M. M. M.","year":"2009","journal-title":"Int. J. Math. Comb."},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20404"},{"key":"atypb30","unstructured":"B. Lidick\u00fd, C. Mattes, and F. Pfender,\n                      $C_5$ Is Almost a Fractalizer\n                      , preprint,https:\/\/arxiv.org\/abs\/2102.06773, 2021."},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2018.07.008"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1007\/BF03322761"},{"key":"atypb33","first-page":"195","volume":"79","author":"Lortz R.","year":"2006","journal-title":"Ars Combin."},{"key":"atypb34","unstructured":"D. Neiman, J. Mackey, and M. Heule,\n                      Tighter Bounds on Directed Ramsey Number $R(7)$\n                      , preprint,https:\/\/arxiv.org\/abs\/2011.00683, 2020."},{"key":"atypb35","unstructured":"S. V. Overberghe,\n                      Algorithms for Computing Ramsey Numbers\n                      , Master's thesis, Ghent University (Advisor: Jan Goedgebeur), Ghent, Belgium, 2020."},{"key":"atypb36","unstructured":"K. Piwakowski,\n                      A new upper bound for $R_3(K_4-e)$\n                      , in Proceedings of the Twenty-eighth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1997), Congr. Numer. 128 (1997), pp. 135-141."},{"key":"atypb37","volume":"1","author":"Radziszowski S. P.","year":"1994","journal-title":"Electron. J. Combin."},{"key":"atypb38","doi-asserted-by":"crossref","unstructured":"F. P. Ramsey,\n                      On a Problem of Formal Logic\n                      , Proc. London Math. Soc., S2-30 (1930), 264,https:\/\/doi.org\/10.1112\/plms\/s2-30.1.264.","DOI":"10.1112\/plms\/s2-30.1.264"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1203350785"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2016.184.3.1"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190020110"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1007\/s003730050025"},{"key":"atypb43","first-page":"70","volume":"45","author":"Shao Z.","year":"2009","journal-title":"Comput. Engrg. Appl."},{"key":"atypb44","first-page":"217","volume":"75","author":"Shao Z.","year":"2010","journal-title":"J. Combin. Math. Combin. Comput."},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1137\/110846622"},{"key":"atypb46","first-page":"189","volume":"62","author":"Tse K.-K.","year":"2007","journal-title":"J. Combin. Math. Combin. Comput."},{"key":"atypb47","unstructured":"E. Vaughan,\n                      Flagmatic 2.0\n                      , 2012,https:\/\/web.archive.org\/web\/20151104132239\/http:\/flagmatic.org\/(discontinued)."},{"key":"atypb48","first-page":"337","volume":"90","author":"Xu X.","year":"2009","journal-title":"Ars Combin."},{"key":"atypb49","unstructured":"M. Yamashita, K. Fujisawa, K. Nakata, M. Nakata, M. Fukuda, K. Kobayashi, and K. Goto,\n                      A High-Performance Software Package for Semidefinite Programs: SDPA 7\n                      ,http:\/\/www.optimization-online.org\/DB_HTML\/2010\/01\/2531.html, 2010."}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1169473","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:06:08Z","timestamp":1787317568000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1169473"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":49,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/18M1169473"],"URL":"https:\/\/doi.org\/10.1137\/18m1169473","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}