Ramsey-type graph coloring and diagonal non-computability

Archive for Mathematical Logic 54 (7-8):899-914 (2015)

Abstract

A function is diagonally non-computable if it diagonalizes against the universal partial computable function. D.n.c. functions play a central role in algorithmic randomness and reverse mathematics. Flood and Towsner asked for which functions h, the principle stating the existence of an h-bounded d.n.c. function implies Ramsey-type weak König’s lemma. In this paper, we prove that for every computable order h, there exists an ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document} -model of h-DNR which is not a not model of the Ramsey-type graph coloring principle for two colors and therefore not a model of RWKL. The proof combines bushy tree forcing and a technique introduced by Lerman, Solomon and Towsner to transform a computable non-reducibility into a separation over ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document} -models.

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References found in this work

RT₂² Does Not Imply WKL₀.Jiayi Liu - 2012 - Journal of Symbolic Logic 77 (2):609-620.
RT2 2 Does Not Imply WKL0.Jiayi Liu - 2012 - Journal of Symbolic Logic 77 (2):609-620.
Lowness for Kurtz Randomness.Noam Greenberg & Joseph S. Miller - 2009 - Journal of Symbolic Logic 74 (2):665-678.

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Citations of this work

The Strength of the Tree Theorem for Pairs in Reverse Mathematics.Ludovic Patey - 2016 - Journal of Symbolic Logic 81 (4):1481-1499.

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