A Completion Result for Partial Affine and Inversive Spaces
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| Format: | Preprint |
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2025
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| _version_ | 1866912727170023424 |
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| author | Grace, Cassie Metsch, Klaus Van de Voorde, Geertrui |
| author_facet | Grace, Cassie Metsch, Klaus Van de Voorde, Geertrui |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_23995 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Completion Result for Partial Affine and Inversive Spaces Grace, Cassie Metsch, Klaus Van de Voorde, Geertrui Combinatorics 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. |
| title | A Completion Result for Partial Affine and Inversive Spaces |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.23995 |