Configuration

Remarks

Clause 4 says that the structure is a partial linear space; it is equivalent to saying that two distinct lines meet in at most one point. Counting incident point-line pairs in two ways gives $p \gamma = \ell \pi$. When $p = \ell$, and hence $\gamma = \pi = k$, the configuration is symmetric and is written $(p_k)$. A geometric configuration is one whose points and lines are points and lines of the real (or another) projective plane, with incidence as in the plane.

Example: the complete quadrilateral, made of $4$ lines in the real plane with no three concurrent and their $6$ intersection points, is a $(6_2\ 4_3)$ configuration. Each line meets the other three lines in $3$ distinct points, each point lies on $2$ of the lines, and $6 \cdot 2 = 4 \cdot 3$.

Non-example: a triangle together with one median has $4$ points and $4$ lines. The vertex the median starts from lies on $3$ of the lines, but each other point lies on only $2$, so clause 2 fails.