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 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.