Classification of ℵ0-categorical C-minimal pure C-sets

We classify all ${\aleph }_0$-categorical and C-minimal C-sets up to elementary equivalence. As usual the Ryll-Nardzewski Theorem makes the classification of indiscernible ${\aleph }_0$-categorical C-minimal sets as a first step. We first define solvable good trees, via a finite induction. The trees involved in initial and induction steps have a set of nodes, either consisting of a singleton, or having dense branches without endpoints and the same number of branches at each node. The class of colored good trees is the elementary class of solvable good trees. We show that a pure C-set M is indiscernible, finite or ${\aleph }_0$-categorical and C-minimal iff its canonical tree $T\left(M\right)$ is a colored good tree. The classification of general ${\aleph }_0$-categorical and C-minimal C-sets is done via finite trees with labeled vertices and edges, where labels are natural numbers, or infinity and complete theories of indiscernible, ${\aleph }_0$-categorical or finite, and C-minimal C-sets.