Ramsey-type graph coloring and diagonal non-computability

Ludovic Patey (1, 2)

1 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
2 PI.R2 - Design, study and implementation of languages for proofs and programs

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.

Mots clés: Reverse mathematics; Forcing; Graph coloring; König's lemma

Domaines: Logique [math.LO]

Ludovic Patey. Ramsey-type graph coloring and diagonal non-computability. Archive for Mathematical Logic, 2015, 54 (7-8), pp.899 - 914. ⟨10.1007/s00153-015-0448-5⟩. ⟨hal-01888596⟩