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

The Ramsey Choice principle for families of $n$-element sets, denoted ${RC}_n$, states that every infinite set $X$ has an infinite subset $Y\subseteq X$ with a choice function on ${\left[Y\right]}^n:=\{z\subseteq Y:|z|=n\}$. We investigate for which positive integers $m$ and $n$ the implication ${RC}_m\Rightarrow {RC}_n$ is provable in $\mathrm{ZF}$. It will turn out that beside the trivial implications ${RC}_m\Rightarrow {RC}_m$, 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 $\mathrm{ZF}$ is ${RC}_2\Rightarrow {RC}_4$.