Switching Sets in PG(3, q)

Abstract

In this note, we are mainly concerned with partial spreads U, V of PG(3, q) which cover the same points and have no line in common. Setting I Ul = I VI =t, we show that if t >q+ then t>max(q+2, 2q-2). Certain applications of this result to (0, 1) matrices and to translation planes are then discussed. 1. Summary. Our purpose is to describe a new result (Theorem 3) on the size of replaceable partial spreads in =PG(3, q), the 3 dimensional projective space over the finite field of order q=pS where p is a prime and s is a positive integer. The bound we obtain is best possible for q odd. It also represents somewhat of a breakthrough in the combinatorial theory of finite translation planes since, when applied to certain classes of such planes, it leads to an improvement of a well-known bound due to R. H. Bruck [1] (see Theorem 4). Furthermore, our result has a purely combinatorial interpretation in terms of (0, 1) matrices; this is discussed in ?5. Although we are dealing with very modern questions, our main tool is a classical theorem in solid geometry, namely the regulus theorem discussed below. By way of an example we shall also come across another hardy perennial-the Schlaffli double-six configuration. The proof in outline of the main result is a pleasant mixture of geometry and combinatorics. II1 order to exhibit this, we have attempted to keep this note accessible and self-contained. Thus we discuss below some geometrical background. We should mention that Theorem 3 can also be stated in purely affine or vector space language. How it can be proved, however, without reverting to the projective situation is far from clear. 2. Background in E. By a partial spread U of E =PG(3, q) we mean any nonempty family of pairwise skew lines of E. If every point P of E lies on some line of U we then refer to U as a full spread or, simply, a spread of E. Two partial spreads U, V of E are said to cover the same points provided the following holds: A point P of E lies on a line of U if and only if P lies on a line of V. A transversal of the partial spread Received by the editors October 20, 1972. AMS (MOS) subject classifications (1970). Primary 50D30, 05B25; Secondary 05B15, 50D30.