Almost-full transversals in equi-$n$-squares
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909423629238272 |
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| author | Chakraborti, Debsoumya Christoph, Micha Hunter, Zach Montgomery, Richard Petrov, Teo |
| author_facet | Chakraborti, Debsoumya Christoph, Micha Hunter, Zach Montgomery, Richard Petrov, Teo |
| contents | In 1975, Stein made a wide generalisation of the Ryser-Brualdi-Stein conjecture on transversals in Latin squares, conjecturing that every equi-$n$-square (an $n\times n$ array filled with $n$ symbols where each symbol appears exactly $n$ times) has a transversal of size $n-1$. That is, it should have a collection of $n-1$ entries that share no row, column, or symbol. In 2017, Aharoni, Berger, Kotlar, and Ziv showed that equi-$n$-squares always have a transversal with size at least $2n/3$. In 2019, Pokrovskiy and Sudakov disproved Stein's conjecture by constructing equi-$n$-squares without a transversal of size $n-\frac{\log n}{42}$, but asked whether Stein's conjecture is approximately true. I.e., does an equi-$n$-square always have a transversal with size $(1-o(1))n$?
We answer this question in the positive. More specifically, we improve both known bounds, showing that there exist equi-$n$-squares with no transversal of size $n-Ω(\sqrt{n})$ and that every equi-$n$-square contains $n-n^{1-Ω(1)}$ disjoint transversals of size $n-n^{1-Ω(1)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07733 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost-full transversals in equi-$n$-squares Chakraborti, Debsoumya Christoph, Micha Hunter, Zach Montgomery, Richard Petrov, Teo Combinatorics In 1975, Stein made a wide generalisation of the Ryser-Brualdi-Stein conjecture on transversals in Latin squares, conjecturing that every equi-$n$-square (an $n\times n$ array filled with $n$ symbols where each symbol appears exactly $n$ times) has a transversal of size $n-1$. That is, it should have a collection of $n-1$ entries that share no row, column, or symbol. In 2017, Aharoni, Berger, Kotlar, and Ziv showed that equi-$n$-squares always have a transversal with size at least $2n/3$. In 2019, Pokrovskiy and Sudakov disproved Stein's conjecture by constructing equi-$n$-squares without a transversal of size $n-\frac{\log n}{42}$, but asked whether Stein's conjecture is approximately true. I.e., does an equi-$n$-square always have a transversal with size $(1-o(1))n$? We answer this question in the positive. More specifically, we improve both known bounds, showing that there exist equi-$n$-squares with no transversal of size $n-Ω(\sqrt{n})$ and that every equi-$n$-square contains $n-n^{1-Ω(1)}$ disjoint transversals of size $n-n^{1-Ω(1)}$. |
| title | Almost-full transversals in equi-$n$-squares |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.07733 |