On extensions of partial geometries containing small \(\mu\)-subgraphs. (Russian) Zbl 0934.51005

The author generalizes the results obtained by D. R. Hughes [Eur. J. Comb. 11, No. 5, 459-471 (1990; Zbl 0719.51005)] and A. Del Fra, D. Ghinelli-Smit, and D. R. Hughes [Geom. Dedicata 42, No. 2, 119-128 (1992; Zbl 0759.51004)].
In this paper there is proved that a geometry, which is a \(\varphi\)-homogeneous extension of partial geometry \(EpG_\alpha(s,t)\), \(\varphi\leq s\) containing \((a,B)\)-antiflag with \(\mu(a,b) =\mu (a,c)=\varphi(1+{t(\varphi-1)\over\alpha})\); \(b,c\in B\), is one of the following geometries: triangular extension or 6-homogeneous extension of generalized quadrangle, extension of a net or dual 2-block design. The \(\varphi\)-homogeneous extensions of partial geometries with strongly regular point graph, where \(\mu\) is given above, are classified.


51E14 Finite partial geometries (general), nets, partial spreads
05B25 Combinatorial aspects of finite geometries