In general, for dimensions $n>5$ we don't have  topological obstructions  to the existence of Einstein metrics (like as Hitchin-Thorpe inequality when $n=4$).

However, there exist   **compact homogeneous Riemannian manifolds** with **no $G$-invariant
Einstein metrics.** In this case we work on cosets $G/K$ of  a compact Lie group $G$ and we consider the Einstein equation $Ric = c \cdot g$  for a $G$-invariant Riemannian metric on $M=G/K$.
For such a metric the Einstein equation reduces to a polynomial system and positive real solutions correspond to invariant Einstein metrics.

Assume now that $\frak{g}=\frak{k}\oplus\frak{m}$ is a reductive decomposition for $G/K$ (such an orthogonal decomposition always exists).  For simplicity,  assume that  tangent space $\frak{m}=T_{o}G/K$ decomposes into $q$ pairwise inequivalent 
isotropy summands (i.e., irreducible $Ad(K)$-submodules)

$
\frak{m}= \frak{m}_1 + \frak{m}_2 + ..... + \frak{m}_q.
$

Then any $G$-invariant Riemannian metric is given by 

$
g = < , > = x_1\cdot (-B)|_{\frak{m}_1}+\cdots+x_s\cdot (-B)|_{\frak{m}_q} 
$

for some positive real numbers $x_1, x_2, ...  x_q$.  Here $-B$ is the negative of the Killing form.

Homogeneous Einstein metrics are real solutions of the system $\{r_1-r_2=0, ...., r_{q-1}-r_{q}=0\}$
where $r_{i}$ are the components of the invariant Ricci tensor on any isotropy summand.
given by 

 \begin{equation}\label{ricc}
   r_{k}=\frac{1}{2x_{k}}+\frac{1}{4d_{k}}\sum_{i, j}\frac{x_{k}}{x_{i}x_{j}}[ijk]-\frac{1}{2d_{k}}\sum_{i, j}\frac{x_{j}}{x_{k}x_{i}}[kij], \qquad (k=1, \ldots, q).
 \end{equation}
Here $[ijk]$ are the **structure constants** of $G/K$.  Their computation is usually non-trivial.  For particular classes of homogeneous spaces (e.g. isotropy irreducible), they are related with the Casimir operator of $\frak{g}$.  They are non-zero only for  these triples $\frak{m}_i$, $\frak{m}_j$, $\frak{m}_k$, for which  

$B([\frak{m}_i, \frak{m}_j], \frak{m}_k)\neq 0$. 


Positive real solutions of the above system may exist or not.  Therefore,  there are  homogeneous spaces with no invariant Einstein metrics. For example, **Wang and Ziller**, by applying the variational approach of homogeneous Einstein metrics on compact homogeneous spaces, they proved that the **12-dimensional** space $SU(4)/SU(2)$ does not admit any homogeneous Einstein metric.

see: M. Wang and W. Ziller: Existence and non-existence of homogeneous Einstein metrics}, 
  Invent.~Math.~84 (1986)  177--194.


 This space seems to be the lowest dimensional example of a compact homogeneous manifold with no invariant Einstein metrics.  In particular, from a recent work of B\"ohm and Kerr we know that :

 **Theorem:**  Any simply connected compact homogeneous Einstein manifold admits at least an invariant Einstein metric.

  B\"ohm and M. Kerr:
   Low-dimensional homogeneous Einstein manifolds,
  Trans. Amer. Math. Soc. 358 (4) (2005) 1455--1468. 

Interesting examples of compact homogeneous spaces with no invariant Einstein metrics were presented also in

  J-S. Park and Y. Sakane:
   Invariant Einstein metrics on certain homogeneous spaces,
  Tokyo J. Math.  20   (1) (1997) 51--61. 

More general arguments about the existence of homogeneous Einstein metrics (which are based on the  topology of compact homogeneous spaces and applications of  variational analysis),   are presented in the articles:

 1)  C. B\"ohm, M. Wang and W. Ziller: A variational approach for homogeneous Einstein metrics}, Geom. Funct. Anal. 14 (2004) (4) 681-733.

2) C. LeBrun and M. Wang (editors):  Surveys in Differential Geometry} Volume VI Essays on Einstein Manifolds, International Press, 1999.

3) C. B\"ohm:  Homogeneous Einstein metrics and simplicial complexes,  J. Diff. Geom. 67 (2004) 79-165.


 We mention that for the problem of **non-homogeneous Einstein  metrics** on homogeneous spaces, **less are known ** (see the work of **Page, or Bohm** for the existence of non homogeneous Einstein metrics.)

For non compact homogeneous manifolds (solvmanifolds, nilmanifolds, etc) we refer the reader to Heber's work and the references therein, although a lot of progress has been made in the last decade in this case too (see the works of Laurent, Tamaru, and others).

J. Heber:  Noncompact homogeneous Einstein space, Invent.~ Math.~133  (1998)   279-352.