Block transitive Steiner systems with more than one point orbit

For all ‘reasonable’ finite t, k and s we construct a t-(א0, k, 1) design and a group of automorphisms which is transitive on blocks and has s orbits on points. In particular, there is a 2-(א0, 4, 1) design with a block-transitive group of automorphisms having two point orbits. This answers a question of P. J. Cameron and C. E. Praeger. The construction is presented in a purely combinatorial way, but is a byproduct of a new way of looking at a model-theoretic construction of E. Hrushovski. 2000 Mathematics Subject Classification: 05B05 (Primary); 20B27 (Secondary) By a t-(v, k, λ) design we mean a set P of points, of cardinality v, together with a set B of blocks each of which is a k-subset of P , and which has the property that any set of t points is a subset of exactly λ blocks. An automorphism of the design (P,B) is simply a permutation of P which preserves B. It is well known that if G is a group of automorphisms of the design (P,B) and t ≥ 2 and v, k, λ are finite, then the number of G-orbits on P is no greater than the number of G-orbits on B: this is commonly referred to as Block’s Lemma ([1]). For the rest of the paper we suppose t ≥ 2, v is infinite and t, k, λ are finite. A standard model-theoretic trick (cf. [2], Section 2.2) shows that the actual infinite cardinality v is irrelevant to our concerns and we will emphasise this by referring to ‘t-(∞, k, λ) designs.’ However, in the designs we construct P will be countably infinite, so v is actually א0. It is known that Block’s Lemma need not hold for t-(∞, k, λ) designs. In unpublished work from 1995, the Author used ideas from [7] to construct a 2-(א0, 4, 14) design admitting a group of automorphisms with one block