Introduction

One of the most natural strictly numerical questions to ask is what can be said if all the line degrees of a linear space S are known. Clearly, this problem will have a reasonable answer only if the set of allowable line degrees is quite small. If there is only one line degree, then S is a design, and in a sense, S is ‘known’. We therefore turn to the case of two line sizes. Work on two consecutive line sizes was the first to appear, and this was done by L. M. Batten, J. Totten and P. de Witte.

If we place an upper bound on v with respect to the line sizes, then we show below that we are able to say something quite precise about S in terms of its structure relative to a projective plane. The case of two line sizes includes the case of one line size. Thus we also give a precise description of designs with v points where v is bounded above. The results of Sections 3.2 and 3.3 are found in de Witte and Batten (1983) and in Batten and Totten (1980).

In Section 3.3 we present the work of Batten (1980) for three consecutive line sizes.

Section 3.4 deals with two non-consecutive line degrees; and in Section 3.5 we briefly describe some general theorems covering a broader class of line size problems.

Let bk and vr be the number of k-lines, respectively r-points, in S.