The Rationals have an AZ-Enumeration

Let M be an -categorical structure (that is, M is countableand Th(M) is -categorical). A nice enumeration of M is a totalordering of M having order-type and satisfying the following.Whenever a_i, i<, is a sequence of elements from M, thereexist some i<j< and an automorphism of M such that (a_i)= a_j and whenever ba_i, then (b)a_j. Such enumerations were introduced by Ahlbrandt and Ziegler in1] where they showed that any Grassmannian of an infinite-dimensionalprojective space over a finite field (or of a disintegratedset) admits a nice enumeration; this combinatorial propertyplayed an essential role in their proof that almost stronglyminimal totally categorical structures are quasi-finitely axiomatisable. Recall that if M is -categorical and is a k-tuple of distinctelements from M (with tp() non-algebraic), then the GrassmannianGr(M; ) is defined as follows. The domain of Gr(M; ) is theset of realisations of tp() in M^k, modulo the equivalence relationxEy if x and y are equal as sets. This is a 0-definable subsetof M^eq, and now the relations on Gr(M; ) are by definition preciselythose which are 0-definable in the structure M^eq. (In particular,Gr(M; ) is also -categorical.) Notice that it is by no means clear that if M admits a niceenumeration, then so do Grassmannians of M. However, there isa strengthening of the notion of nice enumeration for whichthis is the case.