Partial geometry

An incidence structure C=(P,L,I) consists of a set of points, a set of lines, and an incidence relation, or set of flags, I \subseteq P \times L ; a point p is said to be incident with a line l if . It is a (finite) partial geometry if there are integers s,t,\alpha\geq 1 such that: * For any pair of distinct points p and , there is at most one line incident with both of them. * Each line is incident with s+1 points. * Each point is incident with t+1 lines. * If a point p and a line l are not incident, there are exactly \alpha pairs , such that p is incident with m and q is incident with . A partial geometry with these parameters is denoted by .

An incidence structure C=(P,L,I) consists of a set of points, a set of lines, and an incidence relation, or set of flags, I \subseteq P \times L ; a point p is said to be incident with a line l if . It is a (finite) partial geometry if there are integers s,t,\alpha\geq 1 such that: For any pair of distinct points p and , there is at most one line incident with both of them. Each line is incident with s+1 points. Each point is incident with t+1 lines. If a point p and a line l are not incident, there are exactly \alpha pairs , such that p is incident with m and q is incident with . A partial geometry with these parameters is denoted by . A partial linear space S=(P,L,I) of order s, t is called a semipartial geometry if there are integers \alpha\geq 1, \mu such that: If a point p and a line l are not incident, there are either 0 or exactly \alpha pairs , such that p is incident with m and q is incident with . Every pair of non-collinear points have exactly \mu 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 \mathrm{PG}(3, q^2) and only those lines that intersect the plane at infinity in a point of a fixed Baer subplane; it has parameters .

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