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

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 CAC, the ascending-descending sequence principle ADS, and the Cohesive Ramsey Theorem for pairs CRT22. We study these principles over the base theory RCA0⁎, which is weaker than the usual base theory RCA0 considered in reverse mathematics in that it allows only Δ10-induction as opposed to Σ10-induction. In RCA0⁎, 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 Σ10-definable cuts. Because of this, they are all Π30- but not Π11-conservative over RCA0⁎, and, in any model of RCA0⁎+¬RCA0, if they are true then they are computably true relative to some set. The long versions exhibit one of two behaviours: they either imply RCA0 over RCA0⁎ or are Π30-conservative over RCA0⁎. 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 RCA0⁎ and, as a consequence, does not follow from RT22 over RCA0⁎.