Block-transitive t-designs I: point-imprimitive designs

We study block-transitive, point-imprimitive t-(v,k,λ) designs for fixed t, v and k. A simple argument shows that we can assume that such a design admits a maximal imprimitive subgroup of S_v. Delandtsheer and Doyen bounded v in terms of k assuming that t≥2; we obtain stronger bounds assuming that t≥3 or that the design is flag-transitive. We also give a structure theorem for designs which attain the Delandtsheer-Doyen bound for all but a few small values of k, and show that for most values of k, there are exactly three such nonisomorphic designs.