For a mixed hypergraphH= (X,C,D), where C andD are set systems over the vertex set X, a coloring is a partition of X into ‘color classes’such that everyC ∈ Cmeets some class in more than one vertex, and everyD ∈ D has a nonempty intersection with at least two classes.A vertex-order x1, x2, . . . , xn onX (n=|X|) is uniquely colorable if the subhypergraph induced by {xj : 1 j i} has precisely on...

This paper deals with ground states for finite-range interaction potentials in lattice models on S d , where S is a finite set. First, we analyse the projective structure of the set of finite, translation-invariant, measures. Second, we show some drastic differences between d = 1 and d = 2 which shed light on crystalline and quasicrystalline ground states. The latter are modelled as uniquely er...