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 by-product of a new way of looking at a model-theoretic construction of E. Hrushovski.