Restricted linear spaces

By the Fundamental Theorem (see Theorem 1.5.5) we know that b ≧ v for any finite linear space. Moreover, the linear spaces with b = v are precisely the (possibly degenerate) projective planes. It is natural to ask “What are the finite linear spaces satisfying b = v + 1, b = v + 2,…?” Indeed, these questions have been asked and answered in J. Totten (1975, 1976a, b).

In this chapter we want, however, to consider a much more general situation. Following Totten, we call a finite linear space restricted if it satisfies (b − v)2 ≦ v. The aim of this chapter is to classify all restricted linear spaces.

In order to formulate the results one particular notion is important. Let A be an affine plane, and denote by п1,…, пm some of its parallel classes. Denote by S' any linear space with point set {п1,…, пm}. Let S be the linear space

Then S is called an affine plane with the linear space S' at infinity.

For example, the Fano quasi-plane is the affine plane of order 2 with a near pencil at infinity (see Figure 6.1.1). (The affine plane of order 2 consists of the three “vertices” and the “midpoint” of the triangle.)