A Completion Result for Partial Affine and Inversive Spaces
Cassie Grace, Klaus Metsch, Geertrui Van de Voorde
TL;DR
The paper investigates when partial affine planes and inversive spaces can be extended to full structures and provides sharp, Dow-type bounds for completion in the affine case. It develops a general framework for completing certain $2$-designs and translates these results into an enhanced criterion: for large $n$, a partial affine plane with parallelism as an equivalence relation and more than $n^2-\ rac{1}{2}$? No, $n^2-$ lines can be completed to an affine plane, specifically exceeding $n^2-\sqrt{n}$ lines ensures completion; it also tightens the bound via Baer subplane constructions and relates to a bound $g(n)=\min_i g_i(n)$ with $f$ solving $f(f+1)=2n$ for secondary thresholds. The results extend to higher-dimensional designs, showing that certain partial $2$-$(n^d,n,1)$-designs and partial $3$-$(n^d+1,n+1,1)$-designs are completable to full inversive spaces under natural local conditions, thereby advancing embedding and completion problems in finite geometry with constructive insights and tightness examples.
Abstract
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 $n$, $n$ sufficiently large, in which parallelism is an equivalence relation, containing more than $n^2-\sqrt{n}$ lines, can be completed to an affine plane, thus improving the $40$-year old bound of [S. Dow. A completion problem for finite affine planes. Combinatorica, 6:321--325, 1986.] 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 $n^2+n-\sqrt{n}$ lines, can be completed to an inversive plane.