Abstract
A function is diagonally non-computable (d.n.c.) 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 (h-DNR) implies Ramsey-type weak König’s lemma (RWKL). In this paper, we prove that for every computable order h, there exists an \({\omega}\) -model of h-DNR which is not a not model of the Ramsey-type graph coloring principle for two colors (RCOLOR 2) 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 \({\omega}\) -models.
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Patey, L. Ramsey-type graph coloring and diagonal non-computability. Arch. Math. Logic 54, 899–914 (2015). https://doi.org/10.1007/s00153-015-0448-5
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DOI: https://doi.org/10.1007/s00153-015-0448-5