BLOCK-TRANSITIVE T-DESIGNS .1. POINT-IMPRIMITIVE DESIGNS

We study block-transitive, point-imprimitive t-(v, k, lambda) 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). Delandtsh eer and Doyen bounded v in terms of k assuming that t greater-than-or- equal-to 2; we obtain stronger bounds assuming that t greater-than-or- equal-to 3 or that the design is flag-transitive. We also give a struc ture 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.