Using several computer algorithms, we calculate some values and bounds for the function \(e(3,k,n)\), the minimum number of edges in a triangle-free graph on \(n\) vertices with no independent set of size \(k\). As a consequence, the following new upper bounds for the classical two-color Ramsey numbers are obtained:\(R(3,10) \leq 43\), \(\quad\),\(R(3,11) \leq 51\), \(\quad\),\(R(3,12) \leq 60\), \(\quad\),\(R(3,13) \leq 69\) \(\quad\) and,\(R(3,14) \leq 78\).