  We exhibit infinite families of two-dimensional lattices (some of which are
triangulations or quadrangulations of the plane) on which the q-state Potts
antiferromagnet has a finite-temperature phase transition at arbitrarily large
values of q. This unexpected result is proven rigorously by using a Peierls
argument to measure the entropic advantage of sublattice long-range order.
Additional numerical data are obtained using transfer matrices, Monte Carlo
simulation, and a high-precision graph-theoretic method.
