HINDMAN'S THEOREM AND CHOICE

In ZF (i.e. the Zermelo-Fraenkel set theory without the Axiom of Choice (AC)), we investigate the set-theoretic strength of a generalized version of Hindman's theorem and of certain weaker forms of this theorem, which were introduced by Fernandez-Breton [8], with respect to their interrelation with several weak choice principles. In this direction, we determine the status of (this general version of) Hindman's theorem (and of weaker forms) in certain permutation models of ZFA + AC and transfer the results to ZF, strengthen some results of [8] and settle a related open problem from Howard and Rubin [10]; thus filling the gap in information in both [8] and [10].