Implications of Ramsey Choice Principles in ZF

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