Characterizing geometric designs, II

We provide a characterization of the classical point-line designs PG 1 ( 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 PG n − 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 q n − 4 + ⋯ + q + 1 . Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as PG 1 ( n , q ) ; in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.