We present a new condition for the existence of long-range order in
discrete spin systems, which emphasizes the role of entropy and high
dimension. The condition applies to all symmetric nearest-neighbor
discrete spin systems with an interal symmetry of "dominant phases".
Specific applications include a proof of Kotecky's conjecture (1985)
on anti-ferromagnetic Potts models, a strengthening of results of
Lebowitz-Gallavotti (1971) and Runnels-Lebowitz (1975) on
Widom-Rowlinson models and of Burton-Steif (1994) on shifts of finite
type, and a significant extension of results of Engbers-Galvin (2012)
on random graph homomorphisms on the hypercube.
Joint work with Ron Peled.