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

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 CAC, the ascending-descending sequence principle ADS, and the Cohesive Ramsey Theorem for pairs CRT22. We study these principles over the base theory RCA(0)*, which is weaker than the usual base theory RCA(0) considered in reverse mathematics in that it allows only Delta(0)(1)-induction as opposed to Sigma(0)(1)-induction. In RCA(0)*, it may happen that an unbounded subset of N is not in bijective correspondence with 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 RCA(0)*, and, in any model of RCA(0)* + -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 RCA(0) over RCA(0)* or are Pi(0)(3)-conservative over RCA(0)*. The conservation results are obtained using a variant of the so-called grouping principle. We also show that the cohesive set principle COH, a strengthening of CRT22, is never computably true in a model of RCA(0)* and, as a consequence, does not follow from RT22 over RCA(0)*. (C) 2021 Elsevier B.V. All rights reserved.