Definable sets in coloured orders.

A complete theory of linearly ordered structures $T$ satisfies condition (LB), called linear binarity, if every complete type of an increasing sequence of elements of a model of $T$ is uniquely determined by all the 2-types of pairs of consecutive elements; (LB) is shared by all complete theories of coloured orders as shown by Rubin. We introduce a strong form of linear binarity (SLB) and a weak form, called linear finiteness (LF). We prove that convex, parametrically definable subsets of models of theories satisfying (LF) are Boolean combinations of intervals and classes of definable convex equivalence relations; if we allow also unary definable sets in these combinations and assume (SLB), then we get description of all definable sets. Remarkably, we prove that (SLB), up to inter-definability, characterizes theories of coloured orders expanded by arbitrary convex equivalence relations.