An incidence structure consists of points , lines , and flags where a point is said to be incident with a line if . It is a (finite) partial geometry if there are integers such that:
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 .
A partial geometry with these parameters is denoted by .
Properties
The number of points is given by and the number of lines by . The point graph (also known as the collinearity 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
A partial linear space of order is called a semipartial geometry if there are integers such that:
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.
A semipartial geometry is a partial geometry if and only if .
It can be easily shown that the collinearity graph of such a geometry is strongly regular with parameters .
A nice example of such a geometry is obtained by taking the affine points of and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters .