  The Ramsey Choice principle for families of $n$-element sets, denoted
$\mathrm{RC}_n$, states that every infinite set $X$ has an infinite subset
$Y\subseteq X$ with a choice function on $[Y]^n := \{z\subseteq Y : |z| = n\}$.
We investigate for which positive integers $m$ and $n$ the implication
$\mathrm{RC}_m \Rightarrow \mathrm{RC}_n$ is provable in ZF. It will turn out
that beside the trivial implications $\mathrm{RC}_m \Rightarrow \mathrm{RC}_m$,
under the assumption that every odd integer $n>5$ is the sum of three primes
(known as ternary Goldbach conjecture), the only non-trivial implication which
is provable in ZF is $\mathrm{RC}_2 \Rightarrow \mathrm{RC}_4$.
