The free roots of the complete graph
Enrique Casanovas, Frank Olaf Wagner · Proceedings of the American Mathematical Society · 2003
There is a model-completion T n T_n of the theory of a (reflexive) n n -coloured graph ⟨ X , R 1 , … , R n ⟩ \langle X,R_1,\ldots ,R_n\rangle such that R n R_n is total, and R i ∘ R j ⊆ R i + j R_i\circ R_j\subseteq R_{i+j} for all i , j i,j . For n > 2 n>2 , the theory T n T_n is not simple, and does not have the strict order property. The theories T n T_n combine to yield a non-simple theory T ∞ T_\infty without the strict order property, which does not eliminate hyperimaginaries.