Unions of intervals in codes based on powers of sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910579414794240 |
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| author | Karam, Thomas |
| author_facet | Karam, Thomas |
| contents | We prove that for every integer $d \ge 2$ there exists a dense collection of subsets of $[n]^d$ such that no two of them have a symmetric difference that may be written as the $d$th power of a union of at most $\lfloor d/2 \rfloor$ intervals. This provides a limitation on reasonable tightenings of a question of Alon from 2023 and of a conjecture of Gowers from 2009, and investigates a direction analogous to that of recent works of Conlon, Kamčev, Leader, Räty and Spiegel on intervals in the Hales-Jewett theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_15144 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unions of intervals in codes based on powers of sets Karam, Thomas Combinatorics Number Theory 05D05, 05D10, 94B25 We prove that for every integer $d \ge 2$ there exists a dense collection of subsets of $[n]^d$ such that no two of them have a symmetric difference that may be written as the $d$th power of a union of at most $\lfloor d/2 \rfloor$ intervals. This provides a limitation on reasonable tightenings of a question of Alon from 2023 and of a conjecture of Gowers from 2009, and investigates a direction analogous to that of recent works of Conlon, Kamčev, Leader, Räty and Spiegel on intervals in the Hales-Jewett theorem. |
| title | Unions of intervals in codes based on powers of sets |
| topic | Combinatorics Number Theory 05D05, 05D10, 94B25 |
| url | https://arxiv.org/abs/2408.15144 |