Abstract:We give a general construction for Steiner triple systems on a countably infinite point set and show that it yields 2 ℵ 0 nonisomorphic systems all of which are uniform and r-sparse for all finite r ≥ 4. q

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Abstract. It is shown that the countable saturated discrete linear ordering has the small index property, but that the countable 1-transitive linear orders which contain a convex subset isomorphic to ‫ޚ‬ 2 do not. Similar results are also proved in the coloured case.2000 Mathematics Subject Classification. 20B27, 06F15.1. Introduction. In [9] it was shown that the ordered set of rational numbers has the 'small index property' SIP, meaning that any subgroup of its automorphism group having index strictly less than 2 ℵ 0 contains the pointwise stabilizer of a finite set. The small index property has received a great deal of attention in quite a wide variety of special cases. Its model-theoretic significance is that its truth tells us that the natural topological group associated with the structure (under the topology of pointwise convergence) can be recovered from the pure group, and from this one can deduce that the structure is interpretable in the (abstract) automorphism group [5].A conjecture of Macpherson [7] was that the SIP should hold for every ℵ 0 -categorical structure. This was refuted by Hrushovski, but the conjecture remains in modified form (see [6], bottom of page 52).In this paper we look at a class of countable structures which are (mostly) not ℵ 0 -categorical, namely the countable '1-transitive' linear orders classified by Morel [8]. (A linear order X is 1-transitive if its automorphism group acts (singly) transitively on X.) These have arisen as building blocks for various other classes of countable structures, in particular for certain 'cycle-free' partial orders [11], and in [10] the small index property was investigated for some of these structures. Generally the SIP was established there for structures built using only the very simplest of Morel's cases, and it was left open as to whether it might hold more generally. What was wanted was to find for which of Morel's structures the SIP held.Rather disappointingly, we are only able to establish the SIP for ‫ޑ‬ (already known), ‫ޚ‬ (trivial), and ‫ޚ.ޑ‬ (new). For all Morel's other orders, ‫ޚ‬ α and ‫ޚ.ޑ‬ α for ordinals α ≥ 2, the SIP is definitely false. Meanwhile, Duby [3] examined the coloured case, and was able to establish the SIP for all the ℵ 0 -categorical coloured orders among those given in [1, 2] (essentially those which only have finitely many colours, and which contain no discrete orderings in their coding trees). In view of the example of ‫ޚ.ޑ‬ (which we already had shown has the SIP) he asked whether one could establish the SIP for all the saturated structures among those classified, and we answer this affirmatively here.

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This paper contains a classification of countable lower 1-transitive linear orders. This is the first step in the classification of countable 1-transitive trees given in Chicot and Truss (2009): the notion of lower 1-transitivity generalises that of 1-transitivity for linear orders, and it is essential for the structure theory of 1-transitive trees. The classification is given in terms of coding trees, which describe how a linear order is fabricated from simpler pieces using concatenations, lexicographic products and other kinds of construction. We define coding trees and show that a coding tree can be constructed from a lower 1-transitive linear order (X, ) by examining all the invariant partitions on X. Then we show that a lower 1-transitive linear order can be recovered from a coding tree up to isomorphism.