Ramsey-type graph coloring and diagonal non-computability

TL;DR

This paper explores the relationship between diagonally non-computable functions and Ramsey-type graph coloring principles, showing that certain bounded non-computable functions do not imply the Ramsey-type K"onig's lemma.

Contribution

It proves the existence of omega-models of DNR_h that do not satisfy RCOLOR2, separating these principles in reverse mathematics.

Findings

Existence of omega-models of DNR_h not satisfying RCOLOR2

Separation of DNR_h and RWKL in reverse mathematics

Application of bushy tree forcing and non-reducibility techniques

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 (DNR_h) implies the Ramsey-type K\"onig's lemma (RWKL). In this paper, we prove that for every computable order h, there exists an~$\omega$-model of DNR_h which is not a not model of the Ramsey-type graph coloring principle for two colors (RCOLOR2) 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.