In this paper, first we consider the existence and non-existence of Einstein
metrics on the topological 4-manifolds $3\mathbb{CP}^2 # k \bar{\mathbb{CP}}^2$
(for $k \in \{11, 13, 14, 15, 16, 17, 18\}$) by using the idea of \cite{[RS]}
and the constructions in \cite{[PPS]} and in \cite{[PPS1]}. Then, we study the
existence or non-existence of non-singular solutions of the normalized Ricci
flow on the exotic smooth structures of these topological manifolds by
employing the obstruction developed in \cite{[MI]}.