A generalization of a result of Dembowski and Wagner

Abstract

The famous Dembowski-Wagner theorem gives various characterizations of the classical geometric 2-design PG n−1 (n, q) among all 2-designs with the same parameters. One of the characterizations requires that all lines have size q + 1. It was conjectured [2] that this is also true for the designs PG d (n, q) with 2 ≤ d ≤ n − 1. We establish this conjecture, hereby improving various previous results.

Recent results on designs with classical parameters

“…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. In fact, he actually proved Theorem 3.2 under the weaker hypothesis that all lines have size at least q + 1; thus Metsch succeeded in entirely removing the two problematic assumptions made by Lefèvre-Percsy.…”

The characterization problem for designs with the parameters of AGd(n, q)

“…1 In particular, he conjectured that the geometric designs are characterized by their line size q +1 and proved this for d ∈ {2, 3, 4}; for the extreme case d = n − 1, the result in question is a simple consequence of the famous Dembowski-Wagner theorem [8]. A little later, the second author [18] established the validity of the conjecture in general (by actually proving a somewhat stronger, but also more technical result):…”