Strongly $(s-2)$-uniform extensions of partial geometries $pG_\alpha(s,t)$

A geometry of rank $2$ is an incidence system $(P,\mathcal B)$, where $P$ is a set of points and $\mathcal B$ is a family of subsets from $P$, which are called blocks. Two points from $P$ are called collinear if they lie in the same block from $\mathcal B$. A pair $(a,B)$ from $(P,\mathcal B)$ is called a flag if the point $a$ belongs to the block $B$ and an antiflag otherwise. A geometry is called $\varphi$-uniform if, for any antiflag $(a,B)$, the number of points in the block $B$ that are collinear to the point $a$ is either $0$ or $\varphi$; it is called strongly $\varphi$-uniform if this number is always $\varphi$. In this paper, we study strongly $(s-2)$-uniform extensions of partial geometries $pG_\alpha(s,t)$.