Rainbow Ramsey theorem for triples is strictly weaker than the arithmetical comprehension axiom. (English) Zbl 1300.03013

The rainbow Ramsey theorem addresses colorings of \(n\)-tuples. A coloring \(f: [\mathbb N ]^n \to \mathbb N\) is \(k\)-bounded if \(|f^{-1}(c)|\leq k\) for every \(c \in \mathbb N\). A set \(R \subseteq \mathbb N\) is a rainbow for \(f\) if \(f\) is injective on \([R]^n\). For natural numbers \(n\) and \(k\), the rainbow Ramsey theorem states: (RRT\(^n_k\)) If \(f: [\mathbb N ]^n \to \mathbb N\) is \(k\)-bounded, then there is an infinite rainbow for \(f\). Working in the framework of reverse mathematics, the author shows that RCA\(_0\)+RRT\(^3_2\not\vdash\)ACA\(_0\) via an adaptation of cone-avoidance arguments like those used to analyze the strength of Ramsey’s theorem for pairs. This result is sharpened in the author’s article [Ann. Pure Appl. Logic 165, No. 2, 389–408 (2014; Zbl 1300.03012)]. For other results on the strength of the rainbow Ramsey theorem, see the work of B. F. Csima and J. R. Mileti [J. Symb. Log. 74, No. 4, 1310–1324 (2009; Zbl 1188.03044)].


03B30 Foundations of classical theories (including reverse mathematics)
03F35 Second- and higher-order arithmetic and fragments
03D80 Applications of computability and recursion theory
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