Implications of Ramsey Choice Principles in ZF

Abstract

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 ${\mathrm{RC}}_2\Rightarrow {\mathrm{RC}}_4$.

9 pages. arXiv admin note: text overlap with arXiv:2101.07840