Partial linear space


A partial linear space is a basic incidence structure in the field of incidence geometry, that carries slightly less structure than a linear space.
The notion is equivalent to that of a linear hypergraph.

Definition

Let an incidence structure, for which the elements of are called points and the elements of are called lines. S is a partial linear space, if the following axioms hold:
  • any line is incident with at least two points
  • any pair of distinct points is incident with at most one line
If there is a unique line incident with every pair of distinct points, then we get a linear space.

Properties

The De Bruijn–Erdős theorem shows that in any finite linear space which is not a single point or a single line, we have.

Examples