Low-like basis theorems for Ramsey's theorem for pairs in first-order arithmetic
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915723457069056 |
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| author | Ikari, Hiroyuki Yokoyama, Keita |
| author_facet | Ikari, Hiroyuki Yokoyama, Keita |
| contents | We construct an $\ll^2$-solution (also known as a weakly low solution) to ${\mathrm{D}^2}$ within ${\mathrm{B}Σ^0_{3}}$ and prove the $\ll^2$-basis theorem for $\mathrm{RT}^2$ over ${\mathrm{B}Σ^0_{3}}$. The $\ll^2$-basis theorem is a variant of the low basis theorem, which has recently received focus in the context of the first-order part of Ramsey type theorems. For the construction, we use Mathias forcing in an effectively coded $ω$-model of $\mathsf{WKL_0}$ to ensure sufficient computability under the system with weaker induction. Using a similar method, we also show the $\ll^2$-basis theorem for $\mathrm{RT}^2_2$ and $\mathrm{EM}_{<\infty}$, a version of Erdős-Moser principle, within $\mathrm{I}Σ^0_{2}$. These results provide simpler proofs of known results on the $Π^1_1$-conservativities of $\mathrm{RT}^2, \mathrm{RT}^2_2$ and $\mathrm{EM}_{<\infty}$ as corollaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07569 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Low-like basis theorems for Ramsey's theorem for pairs in first-order arithmetic Ikari, Hiroyuki Yokoyama, Keita Logic We construct an $\ll^2$-solution (also known as a weakly low solution) to ${\mathrm{D}^2}$ within ${\mathrm{B}Σ^0_{3}}$ and prove the $\ll^2$-basis theorem for $\mathrm{RT}^2$ over ${\mathrm{B}Σ^0_{3}}$. The $\ll^2$-basis theorem is a variant of the low basis theorem, which has recently received focus in the context of the first-order part of Ramsey type theorems. For the construction, we use Mathias forcing in an effectively coded $ω$-model of $\mathsf{WKL_0}$ to ensure sufficient computability under the system with weaker induction. Using a similar method, we also show the $\ll^2$-basis theorem for $\mathrm{RT}^2_2$ and $\mathrm{EM}_{<\infty}$, a version of Erdős-Moser principle, within $\mathrm{I}Σ^0_{2}$. These results provide simpler proofs of known results on the $Π^1_1$-conservativities of $\mathrm{RT}^2, \mathrm{RT}^2_2$ and $\mathrm{EM}_{<\infty}$ as corollaries. |
| title | Low-like basis theorems for Ramsey's theorem for pairs in first-order arithmetic |
| topic | Logic |
| url | https://arxiv.org/abs/2601.07569 |