We prove the existence of Sasakian-Einstein metrics on infinitely many rational homology spheres in all odd dimensions greater than 3. In dimension 5 we obain somewhat sharper results. There are examples where the number of effective parameters in the Einstein metric grows exponentially with dimension.
"Einstein metrics on rational homology spheres." J. Differential Geom. 74 (3) 353 - 362, November 2006. https://doi.org/10.4310/jdg/1175266230