Homogeneous Einstein Metrics and Butterflies

M.~M.~Graev associated in \cite{Gr} to a compact homogeneous space $G\mathrm{/}H$ a nerve $\text{XGH}$, whose non-contractibility implies the existence of a $G$-invariant Einstein metric on $G\mathrm{/}H$. The nerve $\text{XGH}$ is a compact semi-algebraic set, defined purely Lie theoretically by intermediate subgroups. In this paper we present a detailed description of the work of Graev and the curvature estimates of \cite{Bo}.