Upper Bounds for Some Ramsey Numbers \(R(3,k)\)

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
\(R(3,14) \leq 78\).