{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T09:19:23Z","timestamp":1769851163018,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: $R(3,10) \\le 42$, $R(3,11) \\le 50$, $R(3,13) \\le 68$, $R(3,14) \\le 77$, $R(3,15) \\le 87$, and $R(3,16) \\le 98$. All of them are improvements by one over the previously best known bounds.\u00a0Let $e(3,k,n)$ denote the minimum number of edges in any triangle-free graph on $n$ vertices without independent sets of order $k$. The new upper bounds on $R(3,k)$ are obtained by completing the computation of the exact values of $e(3,k,n)$ for all $n$ with $k \\leq 9$ and for all $n \\leq 33$ for $k = 10$, and by establishing new lower bounds on $e(3,k,n)$ for most of the open cases for $10 \\le k \\le 15$. The enumeration of all graphs witnessing the values of $e(3,k,n)$ is completed for all cases with $k \\le 9$. We prove that the known critical graph for $R(3,9)$ on 35 vertices is unique up to isomorphism. For the case of $R(3,10)$, first we establish that $R(3,10)=43$ if and only if $e(3,10,42)=189$, or equivalently, that if $R(3,10)=43$ then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that $R(3,10) \\le 42$.<\/jats:p>","DOI":"10.37236\/2824","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:59:06Z","timestamp":1578711546000},"source":"Crossref","is-referenced-by-count":3,"title":["New Computational Upper Bounds for Ramsey Numbers $R(3,k)$"],"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":[[2013,2,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i1p30\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i1p30\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:27:10Z","timestamp":1579260430000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i1p30"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,2,5]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2013,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/2824","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,2,5]]},"article-number":"P30"}}