Abstract: This chapter examines substructure lattices, with emphasis on how to obtain models with specific finite lattices as substructure lattices. The finite distributive lattice case is presented in full. The general technique involving canonical partition properties using congruence lattices and theorems such as the Hales-Jewett theorem is described and applied. Wilkie's theorem for the pentagon lattice and Paris' theorem for all countable distributive lattices are also presented.

Keywords: Hales-Jewett theorem, Wilkie's theorem, interstructure lattice, canonical partition property,