  Magnetic frustration and disorder are key ingredients to prevent the onset of
magnetic order. In the disordered hexagonal double perovskite
BaTi$_{1/2}$Mn$_{1/2}$O$_{3}$, Mn$^{4+}$ cations, with $S=3/2$ spins, can
either form highly correlated states of magnetic trimers or dimers or remain as
weakly interacting orphan spins. At low temperature ($T$), the dimer response
is negligible, and magnetism is dominated by the trimers and orphans. To
explore the role of magnetic frustration, disorder and possibly of quantum
fluctuations, the low-$T$ magnetic properties of the remaining magnetic degrees
of freedom of BaTi$_{1/2}$Mn$_{1/2}$O$_{3}$ are investigated. Heat-capacity
data and magnetic susceptibility display no evidence for a phase transition to
a magnetically ordered phase but indicate the formation of a correlated spin
state. The low-temperature spin dynamics of this state is then explored by
$\mu$SR experiments. The zero field $\mu^{+}$ relaxation rate data show no
static magnetism down to $T=19$ mK and longitudinal field experiments support
as well that dynamic magnetism persists at low $T$. Our results are interpreted
in terms of a spin glass state which stems from a disordered lattice of orphans
spins and trimers. A spin liquid state in BaTi$_{1/2}$Mn$_{1/2}$O$_{3}$,
however, is not excluded and is also discussed.
