Characterizing geometric designs, II

Abstract

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 quasisymmetric 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.

Recent results on designs with classical parameters

“…Therefore one would like to find a nicer characterization. In this direction, in 2009, the present author conjectured the following result [27].…”

“…To do so, we consider designs with the parameters of P G 1 (n, q) so that v = q n + · · · + q + 1. In this case, at least the leading term in the bound given by Theorem 3.3 is correct [28]. Proposition 3.4.…”

“…Recently, some further cases were established: the case d = 2 in [30]; the cases d = 3 and d = 4 in [27]; and the case d = n − 2 under an additional assumption on the intersection size of blocks in [28]. Finally, Metsch [39] obtained the general result using an elegant recursive argument.…”

“…We note that the case d = n − 2 treated in [28] is of particular interest, as the classical (v, k, λ)-designs D = P G n−2 (n, q) are actually quasi-symmetric; that is, they have just two intersection numbers, namely x = q n−4 + · · · + q + 1 and y = q n−3 + · · · + q + 1.…”

“…Of course, a geometric characterization of the designs P G 1 (n, q) among all (v, k, 1)-designs (actually among all finite linear spaces) is provided by the classical Veblen-Young axioms for projective spaces; see, for instance, [3,§XII.1]. In [28], we provide the only other known general characterization; it is of a combinatorial nature and uses the notion of a "hyperplane". This needs a bit of preparation.…”