In general, for dimensions $n>5$ we dont have  topological obstrucions  to 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.

Moreover,  if $\frak{g}=\frak{k}\oplus\frak{m}$ is a reductive decomposition for $G/K$, and we 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. (the induced inner product on the tangent space).

Homogeneous Einstein metrics are real soulutions 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, s).
 \end{equation}
Here $[ijk]$ are the structure constants of $G/K$.  Their computation is usually non-trivial.


Positive real solutions of the above system may exist or not.  Therefore,  there are (compact, but also non compact) homogeneous spaces with no invariant Einstein metrics. 
For example, Wang and Ziller, by applying the variational approach to hcompact omogeneous 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-excistence 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 manifod admts 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. 

Other interesting examples of compact homogeneosu spaces with no invariant Einstein metrics were given by Sakane and Park : 

  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 knwon** (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 refernces therin, although a lot of progress has been made in the last decade in this case too.

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