$\mathcal{L}$-intersecting or Configuration Forbidden Families on Set Systems and Vector Spaces over Finite Fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914776410488832 |
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| author | Liu, Jiuqiang Yu, Guihai Feng, Lihua Li, Yongtao |
| author_facet | Liu, Jiuqiang Yu, Guihai Feng, Lihua Li, Yongtao |
| contents | In this paper, we derive a tight upper bound for the size of an intersecting $k$-Sperner family of subspaces of the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over finite field $\mathbb{F}_{q}$ which gives a $q$-analogue of the Erdős' $k$-Sperner Theorem, and we then establish a general relationship between upper bounds for the sizes of families of subsets of $[n] = \{1, 2, \dots, n\}$ with property $P$ and upper bounds for the sizes of families of subspaces of $\mathbb{F}_{q}^{n}$ with property $P$, where $P$ is either $\mathcal{L}$-intersecting or forbidding certain configuration. Applying this relationship, we derive generalizations of the well known results about the famous Erdős matching conjecture and Erdős-Chvátal simplex conjecture to linear lattices. As a consequence, we disprove a related conjecture on families of subspaces of $\mathbb{F}_{q}^{n}$ by Ihringer [Europ. J. Combin., 94 (2021), 103306]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04289 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathcal{L}$-intersecting or Configuration Forbidden Families on Set Systems and Vector Spaces over Finite Fields Liu, Jiuqiang Yu, Guihai Feng, Lihua Li, Yongtao Combinatorics In this paper, we derive a tight upper bound for the size of an intersecting $k$-Sperner family of subspaces of the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over finite field $\mathbb{F}_{q}$ which gives a $q$-analogue of the Erdős' $k$-Sperner Theorem, and we then establish a general relationship between upper bounds for the sizes of families of subsets of $[n] = \{1, 2, \dots, n\}$ with property $P$ and upper bounds for the sizes of families of subspaces of $\mathbb{F}_{q}^{n}$ with property $P$, where $P$ is either $\mathcal{L}$-intersecting or forbidding certain configuration. Applying this relationship, we derive generalizations of the well known results about the famous Erdős matching conjecture and Erdős-Chvátal simplex conjecture to linear lattices. As a consequence, we disprove a related conjecture on families of subspaces of $\mathbb{F}_{q}^{n}$ by Ihringer [Europ. J. Combin., 94 (2021), 103306]. |
| title | $\mathcal{L}$-intersecting or Configuration Forbidden Families on Set Systems and Vector Spaces over Finite Fields |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.04289 |