>k-homogeneous infinite graphs

In this article we give an explicit classification for the countably infinite graphs $\mathcal{G}$ which are, for some $k$, $\geq$$k$-homogeneous. It turns out that a $\geq$$k-$homogeneous graph $\mathcal{M}$ is non-homogeneous if and only if it is either not $1-$homogeneous or not $2-$homogeneous, both cases which may be classified using ramsey theory.