Ramsey-like theorems and immunities
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911668213121024 |
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| author | Mimouni, Ahmed Patey, Ludovic |
| author_facet | Mimouni, Ahmed Patey, Ludovic |
| contents | A Ramsey-like theorem is a statement of the form ``For every 2-coloring of $[\mathbb{N}]^2$, there exists an infinite set~$H \subseteq \mathbb{N}$ such that $[H]^2$ avoids some pattern''. We prove that none of these statements are computably trivial, by constructing a computable 2-coloring of $[\mathbb{N}]^2$ such that every infinite set avoiding any pattern computes a diagonally non-computable function relative to $\emptyset'$. We also consider multiple notions of weaknesses based of variants of immunity, and characterize the Ramsey-like theorems which preserve these notions or not, based on the shape of the avoided pattern. This is part of a larger study of the reverse mathematics of Ramsey-like theorems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ramsey-like theorems and immunities Mimouni, Ahmed Patey, Ludovic Logic 03B30 A Ramsey-like theorem is a statement of the form ``For every 2-coloring of $[\mathbb{N}]^2$, there exists an infinite set~$H \subseteq \mathbb{N}$ such that $[H]^2$ avoids some pattern''. We prove that none of these statements are computably trivial, by constructing a computable 2-coloring of $[\mathbb{N}]^2$ such that every infinite set avoiding any pattern computes a diagonally non-computable function relative to $\emptyset'$. We also consider multiple notions of weaknesses based of variants of immunity, and characterize the Ramsey-like theorems which preserve these notions or not, based on the shape of the avoided pattern. This is part of a larger study of the reverse mathematics of Ramsey-like theorems. |
| title | Ramsey-like theorems and immunities |
| topic | Logic 03B30 |
| url | https://arxiv.org/abs/2508.15597 |