# Partial geometry

An incidence structure (cf. Incidence system) $S = ( P , L , I )$ in which the incidence relation between points and lines is symmetric and satisfies the following axioms:

1) each point is incident to $r$ lines, $r \geq 2$, and two distinct points are incident to at most one line;

2) each line is incident to $k$ points, $k \geq 2$;

3) through each point not incident to a line $l$ there are exactly $t \geq 1$ lines intersecting $l$.

If a partial geometry consists of $v$ points and $b$ lines, then

$$v = \frac{k [ ( k - 1 ) ( r - 1 ) + t ] }{t} \ \textrm{ and } \ \ b = \frac{r [ ( k - 1 ) ( r - 1 ) + t ] }{t} ,$$

and necessary conditions for the existence of such a partial geometry are that $( k - 1 ) ( r - 1 ) k r$ be divisible by $t ( k + r - t - 1 )$, $k ( k - 1 ) ( r - 1 )$ by $t$ and $r ( k - 1 ) ( r - 1 )$ by $t$( cf. [2]).

Partial geometries can be divided into four classes:

a) partial geometries with $t = k$ or (dually) $t = r$. Geometries of this type are just $2 - ( v , k , 1 )$- schemes or $2 - ( v , r , 1 )$- schemes (cf. Block design);

b) partial geometries with $t = k - 1$ or (dually) $t = r - 1$. In this case a partial geometry is the same thing as a net of order $k$ and defect $k - r + 1$( or dually);

c) partial geometries with $t = 1$, known as generalized quadrangles;

d) partial geometries with $1 < t < \min ( k - 1 , r - 1 )$.

#### References

 [1] R.C. Bose, "Strongly regular graphs, partial geometries and partially balanced designs" Pacific J. Math. , 13 : 2 (1963) pp. 389–419 [2] J.A. Thas, "Combinatorics of partial geometries and generalized quadrangles" M. Aigner (ed.) , Higher Combinatorics , Reidel (1977) pp. 183–199 [3] J.A. Thas, "Construction of maximal arcs and partial geometries" Geometrica Dedicata , 3 : 1 (1974) pp. 61–64