On Ramsey Minimal Graphs

A graph $G$ is $r$-Ramsey-minimal with respect to a graph $H$ if every $r$-coloring of the edges of $G$ yields a monochromatic copy of $H$, but the same is not true for any proper subgraph of $G$. In this paper we show that for any integer $k \geq 3$ and $r \geq 2$, there exists a constant $c>1$ such that for large enough $n$, there exist at least $c^{n^2}$ nonisomorphic graphs on at most $n$ vertices, each of which is $r$-Ramsey-minimal with respect to the complete graph $K_k$. Furthermore, in the case $r=2$, we give an asymmetric version of the above result.