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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.04895 |
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| _version_ | 1866910358284795904 |
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| author | Currier, Gabriel Shahriari, Shahriar |
| author_facet | Currier, Gabriel Shahriari, Shahriar |
| contents | Three $k$-dimensional subspaces $A$, $B$, and $C$ of an $n$-dimensional vector space $V$ over a finite field are called a $3$-cluster if $A \cap B \cap C = \{\mathbf{0}_V\}$ and yet $\dim(A+B+C) \leq 2k$. A special kind of $3$-cluster, which we call a covering triple, consists of subspaces $A,B,C$ such that $A = (A \cap B )\oplus (A \cap C)$. We prove that, for $2 \leq k \le n/2$, the largest size of a covering triple-free family of $k$-dimensional subspaces is the same as the size of the largest such star (a family of subspaces all containing a designated non-zero vector). Moreover, we show that if $k < n/2$, then stars are the only families achieving this largest size. This in turn implies the same result for $3$-clusters, which gives the vector space-analogue of a theorem of Mubayi for set systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04895 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $3$-cluster-free families of subspaces Currier, Gabriel Shahriari, Shahriar Combinatorics 05D05 Three $k$-dimensional subspaces $A$, $B$, and $C$ of an $n$-dimensional vector space $V$ over a finite field are called a $3$-cluster if $A \cap B \cap C = \{\mathbf{0}_V\}$ and yet $\dim(A+B+C) \leq 2k$. A special kind of $3$-cluster, which we call a covering triple, consists of subspaces $A,B,C$ such that $A = (A \cap B )\oplus (A \cap C)$. We prove that, for $2 \leq k \le n/2$, the largest size of a covering triple-free family of $k$-dimensional subspaces is the same as the size of the largest such star (a family of subspaces all containing a designated non-zero vector). Moreover, we show that if $k < n/2$, then stars are the only families achieving this largest size. This in turn implies the same result for $3$-clusters, which gives the vector space-analogue of a theorem of Mubayi for set systems. |
| title | $3$-cluster-free families of subspaces |
| topic | Combinatorics 05D05 |
| url | https://arxiv.org/abs/2403.04895 |