  We report studies of the phase diagrams of two two-dimensional systems
composed of particles with repulsive pair potentials, each of which supports a
high-density Kagome lattice phase. The commonalities in the phase diagrams of
the systems we have studied and other 2D systems suggests the existence of a
universal mechanism driving all to favor a similar series of packing
arrangements as the density is increased, but the simulations considered show
that the only such general rule proposed, namely the S\"uto theorem, is not a
necessary condition for the support of multiple distinct lattice structures by
a particular pair potential. We also find that close to the liquid-to-Kagome
phase transition the transient structured fluctuations in the liquid have
Kagome symmetry whereas deeper in the liquid phase the fluctuations have
hexagonal symmetry. As the deviation of the liquid density from the transition
density decreases fluctuations with hexagonal symmetry are replaced with those
with Kagome symmetry. When the transition is string phase-to-Kagome phase the
transient structured fluctuations in the string phase near the transition have
both six-fold and other than six-fold symmetries, with stronger preference for
six-fold symmetry in the Truskett system than in the Torquato system. The path
of the string-to-Kagome transition in the Truskett system involves intermediate
honeycomb configurations that subsequently buckle to form a Kagome lattice. The
path of the string-to-Kagome transition in the Torquato system suggests that
the Kagome phase is formed by coiled strings merging together; increasing
density generates a Kagome phase with imperfections such as 8-particle rings.
