Partial geometry

An incidence structure ${\displaystyle C=(P,L,I)}$ consists of a set ⁠${\displaystyle P}$⁠ of points, a set ⁠${\displaystyle L}$⁠ of lines, and an incidence relation, or set of flags, ${\displaystyle I\subseteq P\times L}$; a point ${\displaystyle p}$ is said to be incident with a line ${\displaystyle l}$ if ⁠${\displaystyle (p,l)\in I}$⁠. It is a (finite) partial geometry if there are integers ${\displaystyle s,t,\alpha \geq 1}$ such that:

• For any pair of distinct points ${\displaystyle p}$ and ⁠${\displaystyle q}$⁠, there is at most one line incident with both of them.
• Each line is incident with ${\displaystyle s+1}$ points.
• Each point is incident with ${\displaystyle t+1}$ lines.
• If a point ${\displaystyle p}$ and a line ${\displaystyle l}$ are not incident, there are exactly ${\displaystyle \alpha }$ pairs ⁠${\displaystyle (q,m)\in I}$⁠, such that ${\displaystyle p}$ is incident with ${\displaystyle m}$ and ${\displaystyle q}$ is incident with ⁠${\displaystyle l}$⁠.

A partial geometry with these parameters is denoted by ⁠${\displaystyle \mathrm {pg} (s,t,\alpha )}$⁠.