A Completion Result for Partial Affine and Inversive Spaces

TL;DR

This paper proves new bounds for completing partial affine and inversive spaces into full structures, extending classical results and applying to higher-dimensional designs with specific parallelism properties.

Contribution

It improves the classical bound for completing partial affine planes and extends the results to higher-dimensional designs and partial inversive spaces.

Findings

Partial affine planes with many lines can be completed to affine planes.

Higher-dimensional design completion results are established.

Conditions for completing partial inversive spaces are derived.

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.