Properties of concepts of freedom and independence for hypergraphs of models of quite o-minimal theories with few countable models

We study properties of the concepts of freedom and independence for hypergraphs of models of a quite o-minimal theory with few countable models. Conditions for freedom of sets of realizations of isolated and non-isolated types are characterized in terms of the convexity rank. In terms of weak orthogonality, characterizations of the relative independence of sets of realizations of isolated and non-isolated types of convexity rank 1 are obtained. Conditions for freedom and independence of equivalence classes are established, indicating the finite rank of convexity of a non-algebraic isolated type of a given theory. In terms of equivalence classes, the conditions for the relative freedom of isolated and non-isolated types are characterized. In terms of weak orthogonality, characterizations of the relative independence of sets of realizations of isolated and non-isolated types over given equivalence relations are obtained. The transfer of the property of relative freedom of types under the action of definable bijections is proved. It is shown that for the specified conditions the non-maximality of the number of countable models of the theory is essential.