Implications of Ramsey Choice principles in ZF$\mathsf {ZF}$

Abstract

AbstractThe Ramsey Choice principle for families of ‐element sets, denoted , states that every infinite set has an infinite subset with a choice function on . We investigate for which positive integers and the implication is provable in . It will turn out that beside the trivial implications , under the assumption that every odd integer is the sum of three primes (known as ternary Goldbach conjecture), the only non‐trivial implication which is provable in is .