The free roots of the complete graph

Abstract

There is a model-completion Tn of the theory of a (reflexive) ncoloured graph (X, R1,...,Rn) such that Rn is total, and Ri ◦ Rj ⊆ Ri+j for all i, j. For n > 2, the theory Tn is not simple, and does not have the strict order property. The theories Tn combine to yield a non-simple theory T∞ without the strict order property, which does not eliminate hyperimaginaries.