A completion result for partial affine and inversive spaces

A partial affine plane of order n is a point-line incidence structure with n 2 points and n points on each line, such that every two lines meet in at most one point. In this paper, we show that a partial affine plane of order n , n ≥ 19 , in which parallelism is an equivalence relation, containing more than n 2 − n lines, can be completed to an affine plane. This bound is tight if n is a square of a prime power. We also show that a partial affine plane of order n , n ≥ 49 , in which parallelism is an equivalence relation and there is no point not lying on any line, containing more than n 2 − f lines, where f ( f + 1 ) = 2 n , can be completed to an affine plane. These results improve on the 40-year old bound of 3 . Furthermore, we derive a higher-dimensional result about the completion of 2- ( n d , n , 1 ) -designs, as well as for partial inversive spaces . In particular, we show that a partial 3- ( n 2 + 1 , n + 1 , 1 ) -design for which in every derived structure, parallelism is an equivalence relation, and there are at least n 2 + n − n lines, can be completed to an inversive plane.