Ramsey-type graph coloring and diagonal non-computability

Ludovic Patey 1, 2
2 PI.R2 - Design, study and implementation of languages for proofs and programs
Inria de Paris, CNRS - Centre National de la Recherche Scientifique, UPD7 - Université Paris Diderot - Paris 7, PPS - Preuves, Programmes et Systèmes
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 ran-domness 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 ω-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 ω-models.
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Ludovic Patey. Ramsey-type graph coloring and diagonal non-computability. Archive for Mathematical Logic, Springer Verlag, 2015, 54 (7-8), pp.899 - 914. ⟨10.1007/s00153-015-0448-5⟩. ⟨hal-01888596⟩

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