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

Abstract

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:
$R(3,10)\le 43$, $$
$R(3,11)\le 51$, $$
$R(3,12)\le 60$, $$
$R(3,13)\le 69$ $$ and
$R(3,14)\le 78$.