Characterizing geometric designs, II

We provide a characterization of the classical point-line designs PG1(n,q), where n⩾3, among all non-symmetric 2-(v,k,1)-designs as those with the maximal number of hyperplanes. As an application of this result, we characterize the classical quasi-symmetric designs PGn−2(n,q), where n⩾4, among all (not necessarily quasi-symmetric) designs with the same parameters as those having line size q+1 and all intersection numbers at least qn−4+⋯+q+1. Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as PG1(n,q); in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.