EpGs with minimal \(\mu\). II (Q1189640)

An extended \(\alpha\)-partial geometry \(\mathcal S\) of order \((s,t)\) is called \(\phi\)-uniform if for each nonincident point-block pair \((P,y)\) of \(\mathcal S\) there are either \(\varphi\) or 0 points incident with \(y\) and collinear with \(P\). A point \(P\) of \(\mathcal S\) is called geometric if for each point \(Q\) at distance 2 from \(P\) the number of points collinear with both \(P\) and \(Q\) attains a certain minimal value. The authors prove that a \(\varphi\)-uniform extended \(\alpha\)-partial geometry with a geometric point and \(t>2\alpha-1\) is a generalized quadrangle, i.e. has \(\alpha=1\). For the case \(t\leq 2\alpha-1\) they have only partial results. [See also the paper above by \textit{A. Del Fra, D. Ghinelli} and the second author, ibid., 119-128 (1992)].