  We report the structural and magnetic properties of the
4}\textit{\textcolor{black}{d }}\textcolor{black}{(M = Rh) based and
5}\textit{\textcolor{black}{d }}\textcolor{black}{(M = Ir) based systems
Ba$_{3}$M$_{x}$Ti$_{3-x}$O$_{9}$ (nominally $x$ = 0.5, 1). The studied
compositions were found to crystallize in a hexagonal structure with the
centrosymmetric space group }$P6_{3}/mmc$.\textcolor{black}{{} The structures
comprise of A$_{2}$O$_{9}$ polyhedra (with the A site (possibly) statistically
occupied by M and Ti) in which pairs of transition metal ions are stacked along
the crystallographic }\textit{\textcolor{black}{c}}\textcolor{black}{-axis.
These pairs form triangular bilayers in the $ab$-plane. The magnetic Rh and Ir
ions occupy these bilayers, diluted by Ti ions even for $x$ = 1. These bilayers
are separated by a triangular layer which is dominantly occupied by Ti ions.
From magnetization measurements we infer strong antiferromagnetic couplings for
all of the materials but the absence of any spin-freezing or spin-ordering down
to 2~K. Further, specific heat measurements down to 0.35~K show no sign of a
phase transition for any of the compounds. Based on these thermodynamic
measurements we propose the emergence of a quantum spin liquid ground state for
Ba$_{3}$Rh$_{0.5}$Ti$_{2.5}$O$_{9}$, and Ba$_{3}$Ir$_{0.5}$Ti$_{2.5}$O$_{9}$,
in addition to the already reported Ba$_{3}$IrTi$_{2}$O$_{9}$. }
