Problem 9: Numerical relationships between lines and spaces

Index

One of the useful characteristics of linear spaces is that we can easily find a relationship between the number of points, v, and the number of lines, b. For instance, except in the trivial case we have seen that b > v in all of our examples of linear spaces. This certainly makes sense as it is necessary to connect every point to every other point via a line. But certainly we can say more about this interesting inequality.

The de Bruijn-Erdos theorem sites some of these relationships:

Let S be a finite linear space with b > 1. Then,

(i) b > v

(ii) if b = v, any two lines have a point in common. In this case, either one line has v - 1 points and all others have two points, or every line has k + 1 points and every point is on k + 1 lines, k > 2.

Let's look at some examples (you will read a proof of the de Bruijn-Erdos theorem later). In (ii), we consider the first case: one line has v - 1 points and all others have 2. This is often referred to as a near-pencil.

For the second case, where every line has k + 1 points and every point is one k + 1 lines, k > 2, we look to the Fano plane as an example (k = 2). In fact, every projective plane satisfies the relationship given in this second case.

We will define a linear space to be point regular when v > 1 and every point is incident with the same number of lines. Similarly, a linear space will be line regular when b > 1 and every line is incident with the same number of points.

We now propose the following lemma:

If a linear space is line regular, then it is point regular (we will prove this later).