  Van der Waerden's (VDW) colouring theorem in combinatoric number theory [1]
has scope for physical applications.The solution of the two colour case has
enabled the construction of an explicit mapping of an infinite, one dimensional
antiferromagnetic Ising system to an effective pseudo- ferromagnetic one with
the coupling constants in the two cases becoming related [2].Here the three
colour problem is solved and the results are used to obtain new insights in the
theory of complex lattices, particularly those relating to ternary alloys. The
existence of these mappings of a multicolour lattice onto a monochromatic one
with different couplings illustrates {\it a new form of duality}.
