Isomorphic Factorization of Complete Bipartite Graph into Forest

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 \(k \geq 4\) and \(n \leq k/2\), 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 \((n, l)\).