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Isomorphisms of linear spaces
Small linear spaces
Block plans
A linear space is an incidence structure which satisfies the following two axioms.

Obviously, a linear space is also a partial linear space. Conversely, given any partial linear space, we may construct a linear space by just joining each pair of non-collinear points with a line of length 2. A proper linear space is a linear space with at least three points on each line. These are the real linear spaces, not arising from a smaller partial linear space.

If is a finite linear space, we denote the number of points and lines of by v and b, respectively. Using this notation, the following important theorem holds.

Theorem. and b=v if and only if is an, eventually degenerated, projective plane.

In fact, all linear spaces with b not too far distant from v are known, in some sense. These are the so called restricted linear spaces.

Contributed by Hauke Klein
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