Abstract

The Ramsey Choice principle for families of n$n$‐element sets, denoted RCn$\operatorname{RC}_{n}$, states that every infinite set X$X$ has an infinite subset Y⊆X$Y\subseteq X$ with a choice function on [Y]n:={z⊆Y:|z|=n}$[Y]^n:= \lbrace z\subseteq Y: |z| = n\rbrace$. We investigate for which positive integers m$m$ and n$n$ the implication RCm⇒RCn$\operatorname{RC}_{m} \implies \operatorname{RC}_{n}$ is provable in ZF$\mathsf {ZF}$. It will turn out that beside the trivial implications RCm⇒RCm$\operatorname{RC}_{m} \implies \operatorname{RC}_{m}$, under the assumption that every odd integer n>5$n>5$ is the sum of three primes (known as ternary Goldbach conjecture), the only non‐trivial implication which is provable in ZF$\mathsf {ZF}$ is RC2⇒RC4$\operatorname{RC}_{2} \implies \operatorname{RC}_{4}$.