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**On the indecomposability of \(\omega^n\).**
*(English)*
Zbl 1260.03017

The authors consider several formulations of pigeonhole principles for finite powers of \(\omega\). The study leads to formalization of two versions of Ramsey’s theorem:

(1) Weak Ramsey’s theorem (WRT\(^2_k\)): For every finite coloring \(c: [\mathbb N ]^2 \to k\) we can find a color \(d\) and an infinite set \(H\) such that for each \(x \in H\) the set \(\{ y \mid c(x,y ) = d \}\) is infinite, and

(2) Hyperweak Ramsey’s theorem (HWRT\(^2_k\)): For every \(c : [\mathbb N ]^2 \to k\) we can find a color \(d\) and an increasing function \(h : \mathbb N \to \mathbb N\) such that, if \(0 < i < j\), there is an \((x,y) \in [h(i-1), h(i) -1 ] \times [h(j-1) , h(j) - 1 ]\) such that \(c(x,y) = d\).

The authors prove that HWRT\(^2_2\) is strictly weaker than WRT\(^2_2\) and that both are weaker than IPT\(^2_2\), as by D. D. Dzhafarov and J. L. Hirst [Arch. Math. Logic 48, No. 2, 141–157 (2009; Zbl 1172.03007)], and stronger than SADS, as in [D. R. Hirschfeldt and R. A. Shore, J. Symb. Log. 72, No. 1, 171–206 (2007; Zbl 1118.03055)]. Other principles introduced in the paper lie between B\(\Pi^0_n\) and I\(\Sigma^0_{n+1}\).

(1) Weak Ramsey’s theorem (WRT\(^2_k\)): For every finite coloring \(c: [\mathbb N ]^2 \to k\) we can find a color \(d\) and an infinite set \(H\) such that for each \(x \in H\) the set \(\{ y \mid c(x,y ) = d \}\) is infinite, and

(2) Hyperweak Ramsey’s theorem (HWRT\(^2_k\)): For every \(c : [\mathbb N ]^2 \to k\) we can find a color \(d\) and an increasing function \(h : \mathbb N \to \mathbb N\) such that, if \(0 < i < j\), there is an \((x,y) \in [h(i-1), h(i) -1 ] \times [h(j-1) , h(j) - 1 ]\) such that \(c(x,y) = d\).

The authors prove that HWRT\(^2_2\) is strictly weaker than WRT\(^2_2\) and that both are weaker than IPT\(^2_2\), as by D. D. Dzhafarov and J. L. Hirst [Arch. Math. Logic 48, No. 2, 141–157 (2009; Zbl 1172.03007)], and stronger than SADS, as in [D. R. Hirschfeldt and R. A. Shore, J. Symb. Log. 72, No. 1, 171–206 (2007; Zbl 1118.03055)]. Other principles introduced in the paper lie between B\(\Pi^0_n\) and I\(\Sigma^0_{n+1}\).

Reviewer: Jeffry L. Hirst (Boone)

### MSC:

03B30 | Foundations of classical theories (including reverse mathematics) |

03F35 | Second- and higher-order arithmetic and fragments |

### Keywords:

reverse mathematics; indecomposability; partition principle; pigeonhole principle; ordinal; Ramsey’s theorem
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\textit{J. R. Corduan} and \textit{F. G. Dorais}, Notre Dame J. Formal Logic 53, No. 3, 373--395 (2012; Zbl 1260.03017)

### References:

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