The strength of Ramsey's theorem for $α$-large sets
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912981247328256 |
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| author | Carlucci, Lorenzo Volpi, Andrea Zdanowski, Konrad |
| author_facet | Carlucci, Lorenzo Volpi, Andrea Zdanowski, Konrad |
| contents | We calibrate the reverse mathematical strength of a family of extensions of Ramsey's theorem to finite colorings of certain subsets of the natural numbers of unbounded finite dimension. Specifically, we analyze the principles $\mathsf{RT}^{!α}_k$ asserting that every $k$-coloring of the exactly $α$-large subsets of an infinite $X \subseteq \mathbb{N}$ admits an infinite homogeneous set, where $α$-largeness is defined via systems of fundamental sequences in the style of Ketonen and Solovay. For each countable ordinal $α< Γ_0$ and each $k \geq 2$, we prove over $\mathsf{RCA}_0$ that the hierarchy of theorems $\mathsf{RT}^{!\a}_k$ corresponds exactly to the hierarchy of systems axiomatized by closure under transfinite Turing jumps, yielding a fine-grained classification between $\mathsf{ACA}_0$ and $\mathsf{ATR}_0$. Our results extend previous work on the case $α=ω$ and provide a uniform correspondence between countable indecomposable ordinals below $Γ_0$ and natural Ramsey-like theorems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22579 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The strength of Ramsey's theorem for $α$-large sets Carlucci, Lorenzo Volpi, Andrea Zdanowski, Konrad Logic Combinatorics 03B30, 05D10, 03F15 We calibrate the reverse mathematical strength of a family of extensions of Ramsey's theorem to finite colorings of certain subsets of the natural numbers of unbounded finite dimension. Specifically, we analyze the principles $\mathsf{RT}^{!α}_k$ asserting that every $k$-coloring of the exactly $α$-large subsets of an infinite $X \subseteq \mathbb{N}$ admits an infinite homogeneous set, where $α$-largeness is defined via systems of fundamental sequences in the style of Ketonen and Solovay. For each countable ordinal $α< Γ_0$ and each $k \geq 2$, we prove over $\mathsf{RCA}_0$ that the hierarchy of theorems $\mathsf{RT}^{!\a}_k$ corresponds exactly to the hierarchy of systems axiomatized by closure under transfinite Turing jumps, yielding a fine-grained classification between $\mathsf{ACA}_0$ and $\mathsf{ATR}_0$. Our results extend previous work on the case $α=ω$ and provide a uniform correspondence between countable indecomposable ordinals below $Γ_0$ and natural Ramsey-like theorems. |
| title | The strength of Ramsey's theorem for $α$-large sets |
| topic | Logic Combinatorics 03B30, 05D10, 03F15 |
| url | https://arxiv.org/abs/2603.22579 |