Partial geometry

• For any pair of distinct points and, there is at most one line incident with both of them.
• Each line is incident with points.
• Each point is incident with lines.
• If a point and a line are not incident, there are exactly pairs, such that is incident with and is incident with.

Properties

• The number of points is given by and the number of lines by.
• The point graph of a is a strongly regular graph:.
• Partial geometries are dual structures: the dual of a is simply a.

  Special case

• The generalized quadrangles are exactly those partial geometries with.
• The Steiner systems are precisely those partial geometries with.

  Generalisations

• If a point and a line are not incident, there are either or exactly pairs, such that is incident with and is incident with.
• Every pair of non-collinear points have exactly common neighbours.