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In Besse's "Einstein manifolds", p. 177, he states that, until that moment, no general classification of homogeneous Einstein manifolds was know, even in the compact case. More specifically, he poses a problem: classify the compact simply connected homogeneous manifolds $M=G/K$ which admit a $G$-invariant Einstein metric.

Does that question remain open to this day?

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Yes, the question is still open.

I suggest to read this quite recent paper by Kerr and Böhm. It reviews some of the most important advances in the problem and includes several open problems.

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