Partial geometry

An <a href="/facts/Incidence_structure/PzqNv9Eq">incidence structure</a> 
 
 
 
 C
 =
 (
 P
 ,
 L
 ,
 I
 )
 
 
 {\displaystyle C=(P,L,I)}
 
 consists of a set ⁠
 
 
 
 P
 
 
 {\displaystyle P}
 
⁠ of points, a set ⁠
 
 
 
 L
 
 
 {\displaystyle L}
 
⁠ of lines, and an incidence relation, or set of flags, 
 
 
 
 I
 ⊆
 P
 ×
 L
 
 
 {\displaystyle I\subseteq P\times L}
 
; a point 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is said to be incident with a line 
 
 
 
 l
 
 
 {\displaystyle l}
 
 if ⁠
 
 
 
 (
 p
 ,
 l
 )
 ∈
 I
 
 
 {\displaystyle (p,l)\in I}
 
⁠. It is a (finite) partial geometry if there are <a href="/facts/Integer/PUwwrawx">integers</a> 
 
 
 
 s
 ,
 t
 ,
 α
 ≥
 1
 
 
 {\displaystyle s,t,\alpha \geq 1}
 
 such that:

<ul><li>For any pair of distinct points 
 
 
 
 p
 
 
 {\displaystyle p}
 
 and ⁠
 
 
 
 q
 
 
 {\displaystyle q}
 
⁠, there is at most one line incident with both of them.</li>
<li>Each line is incident with 
 
 
 
 s
 +
 1
 
 
 {\displaystyle s+1}
 
 points.</li>
<li>Each point is incident with 
 
 
 
 t
 +
 1
 
 
 {\displaystyle t+1}
 
 lines.</li>
<li>If a point 
 
 
 
 p
 
 
 {\displaystyle p}
 
 and a line 
 
 
 
 l
 
 
 {\displaystyle l}
 
 are not incident, there are exactly 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 pairs ⁠
 
 
 
 (
 q
 ,
 m
 )
 ∈
 I
 
 
 {\displaystyle (q,m)\in I}
 
⁠, such that 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is incident with 
 
 
 
 m
 
 
 {\displaystyle m}
 
 and 
 
 
 
 q
 
 
 {\displaystyle q}
 
 is incident with ⁠
 
 
 
 l
 
 
 {\displaystyle l}
 
⁠.</li></ul>
A partial geometry with these parameters is denoted by ⁠
 
 
 
 
 p
 g
 
 (
 s
 ,
 t
 ,
 α
 )
 
 
 {\displaystyle \mathrm {pg} (s,t,\alpha )}
 
⁠.

Partial geometry open-in-new

Partial geometry