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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.08556 |
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| _version_ | 1866910635966595072 |
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| author | Reiher, Christian Rödl, Vojtěch Sales, Marcelo |
| author_facet | Reiher, Christian Rödl, Vojtěch Sales, Marcelo |
| contents | We construct for every integer $k\geq 3$ and every real $μ\in(0, \frac{k-1}{k})$ a set of integers $X=X(k, μ)$ which, when coloured with finitely many colours, contains a monochromatic $k$-term arithmetic progression, whilst every finite $Y\subseteq X$ has a subset $Z\subseteq Y$ of size $|Z|\geq μ|Y|$ that is free of arithmetic progressions of length $k$. This answers a question of Erdős, Nešetřil, and the second author. Moreover, we obtain an analogous multidimensional statement and a Hales-Jewett version of this result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_08556 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Colouring versus density in integers and Hales-Jewett cubes Reiher, Christian Rödl, Vojtěch Sales, Marcelo Combinatorics We construct for every integer $k\geq 3$ and every real $μ\in(0, \frac{k-1}{k})$ a set of integers $X=X(k, μ)$ which, when coloured with finitely many colours, contains a monochromatic $k$-term arithmetic progression, whilst every finite $Y\subseteq X$ has a subset $Z\subseteq Y$ of size $|Z|\geq μ|Y|$ that is free of arithmetic progressions of length $k$. This answers a question of Erdős, Nešetřil, and the second author. Moreover, we obtain an analogous multidimensional statement and a Hales-Jewett version of this result. |
| title | Colouring versus density in integers and Hales-Jewett cubes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2311.08556 |