  We ask a general question: what are locally homogeneous compact
pseudo-Riemannian Einstein manifolds? We show that any standard compact
Clifford-Klein form of a simple non-compact Lie group admits at least one
Einstein metric. We conjecture that these are basically the only possible
locally homogeneous Einstein pseudo-Riemannian compact manifolds using T.
Kobayashi's conjecture as a guiding principle.
