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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2310.17433 |
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| _version_ | 1866914932765753344 |
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| author | Kovács, Benedek |
| author_facet | Kovács, Benedek |
| contents | We consider the following question by Balister, Győri and Schelp: given $2^{n-1}$ nonzero vectors in $\mathbb{F}_2^n$ with zero sum, is it always possible to partition the elements of $\mathbb{F}_2^n$ into pairs such that the difference between the two elements of the $i$-th pair is equal to the $i$-th given vector for every $i$? An analogous question in $\mathbb{F}_p$, which is a case of the so-called "seating couples" problem, has been resolved by Preissmann and Mischler in 2009. In this paper, we prove the conjecture in $\mathbb{F}_2^n$ in the case when the number of distinct values among the given difference vectors is at most $n-2\log n-1$, and also in the case when at least a fraction $\frac12+\varepsilon$ of the given vectors are equal (for all $\varepsilon>0$ and $n$ sufficiently large based on $\varepsilon$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_17433 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finding a perfect matching of $\mathbb{F}_2^n$ with prescribed differences Kovács, Benedek Combinatorics We consider the following question by Balister, Győri and Schelp: given $2^{n-1}$ nonzero vectors in $\mathbb{F}_2^n$ with zero sum, is it always possible to partition the elements of $\mathbb{F}_2^n$ into pairs such that the difference between the two elements of the $i$-th pair is equal to the $i$-th given vector for every $i$? An analogous question in $\mathbb{F}_p$, which is a case of the so-called "seating couples" problem, has been resolved by Preissmann and Mischler in 2009. In this paper, we prove the conjecture in $\mathbb{F}_2^n$ in the case when the number of distinct values among the given difference vectors is at most $n-2\log n-1$, and also in the case when at least a fraction $\frac12+\varepsilon$ of the given vectors are equal (for all $\varepsilon>0$ and $n$ sufficiently large based on $\varepsilon$). |
| title | Finding a perfect matching of $\mathbb{F}_2^n$ with prescribed differences |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2310.17433 |