Divisible semiplanes, arcs, and relative difference sets.

*(English)*Zbl 0637.05011This article initiates the study of arcs and ovals in divisible semiplanes (also called elliptic semiplanes). Connections with relative difference sets are also explained. After giving some general results the author concentrates on divisible semiplanes arising from projective planes by deleting the points and lines of a Baer subplane. Several constructions of arcs in such designs are given, and bounds on the number of points in an arc are proven. In particular, the existence of maximal arcs is investigated. Along the way the following improvement of a result due to E. Johnsen is given: -1 is never a multiplier of an abelian relative difference set. (Johnsen had the additional requirement that the relative difference set be planar.) Finally, the author closes with two interesting conjectures and describes the evidence in their favor.

Reviewer: M.Kallaher

##### MSC:

05B10 | Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) |

05B25 | Combinatorial aspects of finite geometries |

51E20 | Combinatorial structures in finite projective spaces |