Infinite Designs with More Point Orbits than Block Orbits

Abstract

Block’s Lemma states that every automorphism group of a finite $2-(v,k,\lambda )$ design acts with at least as many block orbits as point orbits: this is not the case for infinite designs. Evans constructed a block transitive $2-(v,4,14)$ design with two point orbits using ideas from model theory and Camina generalized this method to construct a family of block transitive designs with two point orbits. In this paper, we generalize the method further to construct designs with $n$ point orbits and $l$ block orbits with $l<n$, where both $n$ and $l$ are finite. In particular, we prove that for $k\ge 4$ and $n\le k/2$, there exists a block transitive $2-(v,k,\lambda )$ design, for some finite $\lambda$, with $n$ point orbits. We also construct $2-(v,4,\lambda )$ designs with automorphism groups acting with $n$ point orbits and $l$ block orbits, $l<n$, for every permissible pair $(n,l)$.