Weaker cousins of Ramsey's theorem over a weak base theory

TL;DR

This paper investigates weaker forms of Ramsey's theorem principles within a limited base theory, revealing their logical strength, conservation properties, and relationships with other combinatorial principles in reverse mathematics.

Contribution

It introduces and analyzes the 'normal' and 'long' variants of Ramsey-related principles over a weaker base theory, establishing their logical equivalences and conservation results.

Findings

Normal principles are equivalent to their relativizations to $oldsymbol{ ext{Sigma}^0_1}$-definable cuts.

Long principles either imply $oldsymbol{ ext{RCA}_0}$ or are $oldsymbol{ ext{Pi}^0_3}$-conservative over $oldsymbol{ ext{RCA}^*_0}$.

Cohesion principle $oldsymbol{ ext{COH}}$ is not computably true in models of $oldsymbol{ ext{RCA}^*_0}$ and does not follow from $oldsymbol{ ext{RT}^2_2}$.

Abstract

The paper is devoted to a reverse-mathematical study of some well-known consequences of Ramsey's theorem for pairs, focused on the chain-antichain principle $\mathsf{CAC}$, the ascending-descending sequence principle $\mathsf{ADS}$, and the Cohesive Ramsey Theorem for pairs $\mathsf{CRT}^2_2$. We study these principles over the base theory $\mathsf{RCA}^*_0$, which is weaker than the usual base theory $\mathsf{RCA}_0$ considered in reverse mathematics in that it allows only $\Delta^0_1$-induction as opposed to $\Sigma^0_1$-induction. In $\mathsf{RCA}^*_0$, it may happen that an unbounded subset of $\mathbb{N}$ is not in bijective correspondence with $\mathbb{N}$. Accordingly, Ramsey-theoretic principles split into at least two variants, "normal" and "long", depending on the sense in which the set witnessing the principle is required to be infinite. We prove that the normal versions of our principles, like that of Ramsey's theorem for pairs and two colours, are equivalent to their relativizations to proper $\Sigma^0_1$-definable cuts. Because of this, they are all $\Pi^0_3$- but not $\Pi^1_1$-conservative over $\mathsf{RCA}^*_0$, and, in any model of $\mathsf{RCA}^*_0 + \neg \mathsf{RCA}_0$, if they are true then they are computably true relative to some set. The long versions exhibit one of two behaviours: they either imply $\mathsf{RCA}_0$ over $\mathsf{RCA}^*_0$ or are $\Pi^0_3$-conservative over $\mathsf{RCA}^*_0$. The conservation results are obtained using a variant of the so-called grouping principle. We also show that the cohesion principle $\mathsf{COH}$, a strengthening of $\mathsf{CRT}^2_2$, is never computably true in a model of $\mathsf{RCA}^*_0$ and, as a consequence, does not follow from $\mathsf{RT}^2_2$ over $\mathsf{RCA}^*_0$.