A note on block-transitive point-imprimitive designs

We study block-transitive point-imprimitive designs. Cameron and Praeger showed that in such designs t=2 or 3. In 2000, Mann and Tuan showed that if t=3 then v≤k2+1. In this paper, we show a finiteness theorem for block-transitive point-imprimitive 3-designs for which this bound is met. We prove that for a given natural number n, there are only finitely many numbers k of the form npα−1, where p is a prime number, such that there exists a block-transitive point-imprimitive 3−k2+1,k,λ design.