  We show that noncompact homogeneous spaces not diffeomorphic to Euclidean
space of dimension 9 or 10 admit no homogeneous Einstein metrics of negative
Ricci curvature, with only three potential exceptions. The main ingredient in
the proof is to show, via a cohomogeneity-one approach, that noncompact
homogeneous spaces admitting an ideal isomorphic to $
\mathfrak{sl}_2(\mathbb{R}) $ admit no homogeneous Einstein metrics of negative
Ricci curvature.
