Extended partial geometries with minimal \(\mu\) (Q1189639)

An extended \(\alpha\)-partial geometry of order \((s,t)\) is a point-block incidence structure \(\mathcal S\) such that the residue \({\mathcal S}_ P\) at any point \(P\) of \(\mathcal S\) is an \(\alpha\)-partial geometry of order (\(s,t\)). For two points \(P\), \(Q\) at distance 2 in the point graph of \(\mathcal S\) let \({\mathcal S}_{P,Q}\) denote the set of points collinear with both \(P\) and \(Q\). The authors find two lower bounds for the number \(\mu(P,Q)=| {\mathcal S}_{P,Q}|\) and they characterize those extended partial geometries in which either of these bounds is attained. Furthermore, they examine the case of locally triangular extended partial geometries more closely and prove some characterization theorems.