Uniform extensions of partial geometries

A geometry of rank 2 is an incidence system $(P,\mathcal B)$, where $P$ is a set of points and $\mathcal B$ is a set of subsets from $P$, called blocks. Two points are called collinear if they lie in a common block. A pair $(a,B)$ from $(P,\mathcal B)$ is called a flag if its point belongs to the block, and an antiflag otherwise. A geometry is called $\varphi$-uniform ($\varphi$ is a natural number) if for any antiflag $(a,B)$ the number of points in the block $B$ collinear to the point a equals 0 or$\varphi$, and strongly $\varphi$-uniform if this number equals $\varphi$. In this paper, we study $\varphi$-uniform extensions of partial geometries $pG_\alpha(s,t)$ with $\varphi=s$ and strongly $\varphi$-uniform geometries with $\varphi=s-1$. In particular, the results on extensions of generalized quadrangles, obtained earlier by Cameron and Fisher, are extended to the case of partial geometries.