Partial linear space

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A partial linear space (also semilinear or near-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.


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.


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



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