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Beschreibung:
We provide a characterization of the classical geometric designs formed by the points and lines of the projective space PG(n,q) of dimension n over the field with q elements, where n >= 3, among all non-symmetric (v,k,1)-designs as those with the maximal number of hyperplanes. As an application of this result, we also characterize the classical quasi-symmetric designs formed by the points and (n-2)-dimensional subspaces of PG(n,q), where n >= 4, among all (not necessarily quasi-symmetric) designs with the same parameters as those having line size q+1 and sufficiently large intersection numbers. Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as the classical point-line designs; in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.