On ramsey’s theorem and the existence of infinite chains or infinite anti-chains in infinite posets

Abstract

Ramsey’s Theorem is naturally connected to the statement “every infinite partially ordered set has either an infinite chain or an infinite anti-chain”. Indeed, it is a well-known result that Ramsey’s Theorem implies the latter principle. In the book “Consequences of the Axiom of Choice” by P. Howard and J. E. Rubin, it is stated as unknown whether the above implication is reversible, that is whether the principle “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” implies Ramsey’s Theorem. The purpose of this paper is to settle the aforementioned open problem. In particular, we construct a suitable Fraenkel- Mostowski permutation model N for ZFA and prove that the above principle for infinite partially ordered sets is true in N, whereas Ramsey’s Theorem is false in N. Then, based on the existence of N and on results of D. Pincus, we show that there is a model of ZF which satisfies “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” and the negation of Ramsey’s Theorem. In addition, we prove that Ramsey’s Theorem (hence, the above principle for infinite partially ordered sets) is true inMostowski’s linearly ordered model, filling the gap of information in the book “Consequences of the Axiom of Choice”.