Confluence Graphs of Unitals
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911125849767936 |
|---|---|
| author | Grundhöfer, Theo Stroppel, Markus J. Van Maldeghem, Hendrik |
| author_facet | Grundhöfer, Theo Stroppel, Markus J. Van Maldeghem, Hendrik |
| contents | We show that the cliques of maximal size in the confluence graph of an arbitrary unital of order $q>2$ have size $q^2$, and that these cliques are the pencils of all blocks through a given point. This solves the Erdős-Ko-Rado problem for all unitals. We also determine all maximal cliques of the confluence graph of the Hermitian unitals. As an application, we show that the confluence graph of an arbitrary unital unambiguously determines the unital. Along the way, we show that each linear space with $q^2$ points such that the sizes of both point rows and line pencils are bounded above by $q+1$ embeds in a projective plane of order $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11693 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Confluence Graphs of Unitals Grundhöfer, Theo Stroppel, Markus J. Van Maldeghem, Hendrik Combinatorics 05B05 05B25 05E30 51A45 51E10 We show that the cliques of maximal size in the confluence graph of an arbitrary unital of order $q>2$ have size $q^2$, and that these cliques are the pencils of all blocks through a given point. This solves the Erdős-Ko-Rado problem for all unitals. We also determine all maximal cliques of the confluence graph of the Hermitian unitals. As an application, we show that the confluence graph of an arbitrary unital unambiguously determines the unital. Along the way, we show that each linear space with $q^2$ points such that the sizes of both point rows and line pencils are bounded above by $q+1$ embeds in a projective plane of order $q$. |
| title | Confluence Graphs of Unitals |
| topic | Combinatorics 05B05 05B25 05E30 51A45 51E10 |
| url | https://arxiv.org/abs/2311.11693 |