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$_{9}$. }
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