Minimal linear spaces

A decent linear space (DLS) is a linear space (or PBD of index 1) without lines of size 1 or 2; see Beth, Jungnickel, and Lenz [l] for background and definitions. We denote the maximal line size of a DLS by k and write DLS(k), then; if we also want to specify the number u of points, we use the notation DLS(k; v). We always assume all linear spaces to be non-trivial, i.e., u # k. We shall be concerned with the “smallest examples” of linear spaces with given k. Denote by uk (resp. bk) the smallest number of points (resp. lines) in a DLS(k). We will determine these numbers for all choices of k. In particular, we shall prove that any DLS(k) with b, lines has uk points. (Note that this is not clear, a priori: If u is too small, one might be forced to use too many small lines.) This result justifies calling a DLS(k) a minimal linear space if it has exactly b, lines. We shall now recall a few auxiliary notions and results which will be needed. We will in fact also require linear spaces with lines of size 2; so these are indecent linear spaces (ILS). Let D be a resolvable ILS on u 229 0097-3165/87 53.00