Covering points with planes
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913689237454848 |
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| author | Dao, Hailong Dhar, Manik Łaba, Izabella Lund, Ben |
| author_facet | Dao, Hailong Dhar, Manik Łaba, Izabella Lund, Ben |
| contents | Suppose that each proper subset of a set $S$ of points in a vector space is contained in the union of planes of specified dimensions, but $S$ itself is not contained in any such union. How large can $|S|$ be?
We prove a general upper bound on $|S|$, which is tight in some cases, for example when all of the planes have the same dimension. We produce an example showing that this upper bound does not hold for point sets whose proper subsets are covered by lines in $(\mathbb{Z}/p^k\mathbb{Z})^2$ with $k\geq 2$, and prove an upper bound in this case. We also investigate the analogous problem for general matroids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08945 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Covering points with planes Dao, Hailong Dhar, Manik Łaba, Izabella Lund, Ben Combinatorics Suppose that each proper subset of a set $S$ of points in a vector space is contained in the union of planes of specified dimensions, but $S$ itself is not contained in any such union. How large can $|S|$ be? We prove a general upper bound on $|S|$, which is tight in some cases, for example when all of the planes have the same dimension. We produce an example showing that this upper bound does not hold for point sets whose proper subsets are covered by lines in $(\mathbb{Z}/p^k\mathbb{Z})^2$ with $k\geq 2$, and prove an upper bound in this case. We also investigate the analogous problem for general matroids. |
| title | Covering points with planes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.08945 |