Positive Einstein metrics on infinitely many four-manifolds

We construct smooth Einstein metrics with positive scalar curvature on $\#_k({\mathbb S^2}\times {\mathbb S^2})$ and on $k\mathbb{CP}^2\#(2k+1)\overline{\mathbb{CP}}{}^{2}$, for $k\geqslant2$. In particular, we show that positive Einstein metrics exist on infinitely many homeomorphism types of close