Cohesive avoidance and arithmetical sets

TL;DR

This paper demonstrates that the stable Ramsey's theorem for pairs does not imply the cohesive principle in certain models, by constructing sequences of sets where no arithmetical partition yields a cohesive set computable by an infinite subset.

Contribution

It shows, via a forcing argument, that for any finite coloring, there exists an infinite homogeneous set that does not compute a given non-computable set, challenging previous assumptions.

Findings

Cohesive principle is not implied by the stable Ramsey's theorem in ω-models.

Constructs sequences of sets with no arithmetical partition yielding a cohesive set.

Uses a forcing method adapted from Seetapun's technique.

Abstract

An open question in reverse mathematics is whether the cohesive principle, $\COH$, is implied by the stable form of Ramsey's theorem for pairs, $\SRT^2_2$, in $\omega$-models of $\RCA$. One typical way of establishing this implication would be to show that for every sequence $\vec{R}$ of subsets of $\omega$, there is a set $A$ that is $\Delta^0_2$ in $\vec{R}$ such that every infinite subset of $A$ or $\bar{A}$ computes an $\vec{R}$-cohesive set. In this article, this is shown to be false, even under far less stringent assumptions: for all natural numbers $n \geq 2$ and $m < 2^n$, there is a sequence $\vec{R} = \sequence{R_0,...,R_{n-1}}$ of subsets of $\omega$ such that for any partition $A_0,...,A_{m-1}$ of $\omega$ arithmetical in $\vec{R}$, there is an infinite subset of some $A_j$ that computes no set cohesive for $\vec{R}$. This complements a number of previous results in computability theory on the computational feebleness of infinite sets of numbers with prescribed combinatorial properties. The proof is a forcing argument using an adaptation of the method of Seetapun showing that every finite coloring of pairs of integers has an infinite homogeneous set not computing a given non-computable set.