Groups of order-automorphisms of the rationals with prescribed scale type

It is shown that, for any given m and n with m < n (where n = ∞ is allowed), there is a group of order-preserving permutations of the rational numbers whose degrees of homogeneity and uniqueness are m and n, respectively. This is in contrast with the situation for the real numbers. The result is deduced from a more general theorem applying to a wide class of relational structures. The tools used are Fraïssé's theorem on homogeneous structures and a lemma of Tits. All the groups constructed are isomorphic to the free group of countable rank.