Reverse mathematics and Ramsey’s property for trees.(English)Zbl 1203.03018

The authors explore the reverse mathematics of various formulations of Ramsey’s theorem on partial orderings. A partial ordering $$P$$ satisfies Ramsey’s theorem for $$n$$-tuples, denoted by RT$$^n$$($$P$$), if for every finite coloring of the $$n$$-element chains of $$P$$ there is a monochromatic subordering which is order-isomorphic to $$P$$. The restriction of RT$$^n$$($$P$$) to $$k$$-colorings is denoted by RT$$^n_k$$($$P$$). Working in RCA$$_0$$, the authors show that when $$n \geq 3$$ and $$k\geq 2$$, ACA$$_0$$ is equivalent to the statement “RT$$^n_k$$($$P$$) holds if and only if the full binary tree $$2^{<\omega }$$ can be embedded in $$P$$.” They also describe a family of trees for which Ramsey’s theorem for pairs is equivalent to ACA$$_0$$. In the last section, they consider RT$$^1$$($$2^{<\omega}$$), which is a pigeonhole principle on $$2^\omega$$, proving that if $$T$$ is an extension of RCA$$_0$$ by $$\Pi^1_1$$ axioms, then $$T$$ proves RT$$^1$$($$2^{<\omega}$$) if and only if $$T$$ proves the induction scheme for $$\Sigma^0_2$$ formulas. In particular, RT$$^1$$($$2^{<\omega}$$) is strictly stronger than the usual infinite pigeonhole principle. This work addresses questions posed by J. Chubb, J. L. Hirst, and T. H. McNicholl [J. Symb. Log. 74, No. 1, 201–215 (2009; Zbl 1162.03009)].

MSC:

 03B30 Foundations of classical theories (including reverse mathematics) 03F35 Second- and higher-order arithmetic and fragments

Zbl 1162.03009
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References:

 [1] Subsystems of second order arithmetic (1999) · Zbl 0909.03048 [2] On the strength of Ramsey’s theorem for pairs 66 pp 1– (2001) · Zbl 0977.03033 [3] {$$\Sigma$$}2 induction and infinite injury priority argument, I. Maximal sets and the jump operator 63 pp 797– (1998) [4] Reverse mathematics, computability, and partitions of trees 74 pp 201– (2009) · Zbl 1162.03009
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