{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T04:39:18Z","timestamp":1769834358973,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Using computer algorithms we establish that\u00a0the Ramsey number $R(3,K_{10}-e)$ is equal to 37, which\u00a0solves the smallest open case for Ramsey numbers of this type.\u00a0We also obtain new upper bounds for the cases of $R(3,K_k-e)$\u00a0for $11 \\le k \\le 16$, and show by construction a new lower\u00a0bound $55 \\le R(3,K_{13}-e)$.The new upper bounds on $R(3,K_k-e)$ are\u00a0obtained by using the values and lower bounds on $e(3,K_l-e,n)$\u00a0for $l \\le k$, where\u00a0$e(3,K_k-e,n)$ is the minimum number of edges in any\u00a0triangle-free graph on $n$ vertices without $K_k-e$ in\u00a0the complement.\u00a0We complete the computation of the exact\u00a0values of $e(3,K_k-e,n)$ for all $n$ with $k \\leq 10$ and for\u00a0$n \\leq 34$ with $k = 11$, and establish many new lower\u00a0bounds on $e(3,K_k-e,n)$ for higher values of $k$.Using the maximum triangle-free graph generation method,\u00a0we determine two other previously unknown Ramsey numbers,\u00a0namely $R(3,K_{10}-K_3-e)=31$ and $R(3,K_{10}-P_3-e)=31$.\u00a0For graphs $G$ on 10 vertices,\u00a0besides $G=K_{10}$, this leaves 6 open\u00a0cases of the form $R(3,G)$. The hardest among them\u00a0appears to be $G=K_{10}-2K_2$, for which we establish the\u00a0bounds $31 \\le R(3,K_{10}-2K_2) \\le 33$.<\/jats:p>","DOI":"10.37236\/3684","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:12:42Z","timestamp":1578687162000},"source":"Crossref","is-referenced-by-count":4,"title":["The Ramsey Number $R(3,K_{10}-e)$ and Computational Bounds for $R(3,G)$"],"prefix":"10.37236","volume":"20","author":[{"given":"Jan","family":"Goedgebeur","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stanis\u0142aw P.","family":"Radziszowski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2013,11,15]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i4p19\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i4p19\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T06:11:02Z","timestamp":1579241462000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i4p19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,11,15]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2013,10,14]]}},"URL":"https:\/\/doi.org\/10.37236\/3684","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,11,15]]},"article-number":"P19"}}