The free roots of the complete graph

Abstract:

There is a model-completion $T_n$ of the theory of a (reflexive) $n$-coloured graph $\langle X,R_1,\ldots ,R_n\rangle$ such that $R_n$ is total, and $R_i\circ R_j\subseteq R_{i+j}$ for all $i,j$. For $n>2$, the theory $T_n$ is not simple, and does not have the strict order property. The theories $T_n$ combine to yield a non-simple theory $T_\infty$ without the strict order property, which does not eliminate hyperimaginaries.