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Thin set theorems and cone avoidance. (English) Zbl 1442.03007
The authors explore the logical, computational, and combinatorial strength of a thin set theorem, $$\mathsf{RT}^n_{<\infty ,\ell}$$ that asserts that for every finite coloring of $$[\mathbb N ]^n$$ (the subsets of $$\mathbb N$$ of size $$n$$) there is an infinite set $$H$$ such that $$[ H ]^n$$ contains at most $$\ell$$ colors. The theorem can be viewed as a problem, where each coloring instance has a solution set $$H$$. A set $$A$$ is $$\mathsf{RT}^n_{<\infty ,\ell}$$-encodable if there is an instance such that every solution computes $$A$$.
Among the results of the paper is the following three-part complete characterization of the $$\mathsf{RT}^n_{<\infty ,\ell}$$-encodable sets, involving an unexpected appearance of $$d_n$$, the $$n^{\text{th}}$$ Catalan number.
(1) The $$\mathsf{RT}^n_{<\infty ,\ell}$$-encodable sets are the hyperarithmetic sets if and only if $$\ell<2^{n-1}$$.
(2) The $$\mathsf{RT}^n_{<\infty ,\ell}$$-encodable sets are the arithmetic sets if and only if $$2^{n-1}\le \ell < d_n$$.
(3) The $$\mathsf{RT}^n_{<\infty ,\ell}$$-encodable sets are the computable sets if and only if $$d_n \le \ell$$.
The authors also show that $$\mathsf{RT}^n_{<\infty ,\ell}$$ admits strong cone avoidance if and only if $$\ell \ge d_n$$, and admits cone avoidance if and only if $$l \ge d_{n-1}$$. Establishing these thresholds answers questions arising from the work of W. Wang [Adv. Math. 261, 1–25 (2014; Zbl 1307.03011)] on an achromatic Ramsey theorem, F. G. Dorais et al. [Trans. Am. Math. Soc. 368, No. 2, 1321–1359 (2016; Zbl 06560459)], and A. Montalbán [Bull. Symb. Log. 17, No. 3, 431–454 (2011; Zbl 1233.03023)]. The results extend the program of the second author [Computability 6, No. 3, 209–221 (2017; Zbl 1420.03028); Isr. J. Math. 216, No. 2, 905–955 (2016; Zbl 1368.03018)], and are used in recent work of R. Downey et al. [“Relationships between computability-theoretic properties of problems”, Preprint, arXiv:1903.04273] and the second author [“Ramsey-like theorems and moduli of computation”, Preprint, arXiv:1901.04388].
##### MSC:
 03B30 Foundations of classical theories (including reverse mathematics) 03F35 Second- and higher-order arithmetic and fragments 03D80 Applications of computability and recursion theory 05D10 Ramsey theory
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