  We report the first observation of homogeneous crystallization in simulated
high-dimensional ($d > 3$) liquids that follow physically realistic dynamics
and have system sizes that are large enough to eliminate the possibility that
crystallization was induced by the periodic boundary conditions. Supercooled
four-dimensional (4D) Lennard-Jones liquids maintained at zero pressure and
constant temperatures $0.59 < T < 0.63$ crystallized within $\sim 2 \times
10^4\tau$, where $\tau$ is the LJ time unit. WCA liquids that were maintained
at the same densities and temperatures at which their LJ counterparts nucleated
did not crystallize even after $2.5\times 10^5\tau$, showing that the presence
of long-ranged attractive interactions dramatically speeds up 4D
crystallization, much as it does in 3D. However, the overlap of the liquid and
crystalline phases' local-bond-order distributions is much smaller for LJ than
for WCA systems, which is the opposite of the 3D trend. This implies that the
widely accepted hypothesis that increasing geometrical frustration rapidly
suppresses crystallization as the spatial dimension $d$ increases is only
generally valid in the absence of attractive interparticle forces.
