This past summer, I read a paper which described that the Ramsey number of a complete graph on $p$ nodes and any graph on $n$ nodes is $N = R(K_p, G_n) \geq (n-1)(p-1) + 1$. I know that the proof of this was really cool! Essentially, you assume that $N = (n-1)(p-1)$ and show that there exists a special Ramsey coloring called the Turan construction. The Ramsey coloring is where you simply make $p-1$ copies of red $K_{n-1}$ and color every other remaining edge in $K_N$ blue. Then there is no red $G_n$ and no blue $K_p$.
I lost my original source for this proof. Does anyone know of a source for this? Its such a cool result!