On Ramsey choice and partial choice for infinite families of n-element sets

Abstract

For an integer $n\ge 2$, Ramsey Choice$\mathsf {RC}_{n}$ is the weak choice principle “every infinite setxhas an infinite subset y such that$[y]^{n}$ (the set of alln-element subsets of y) has a choice function”, and $\mathsf {C}_{n}^{-}$ is the weak choice principle “every infinite family of n-element sets has an infinite subfamily with a choice function”. In 1995, Montenegro showed that for $n=2,3,4$, $\mathsf {RC}_{n}\rightarrow \mathsf {C}_{n}^{-}$. However, the question of whether or not $\mathsf {RC}_{n}\rightarrow \mathsf {C}_{n}^{-}$ for $n\ge 5$ is still open. In general, for distinct $m,n\ge 2$, not even the status of “$\mathsf {RC}_{n}\rightarrow \mathsf {C}_{m}^{-}$” or “$\mathsf {RC}_{n}\rightarrow \mathsf {RC}_{m}$” is known. In this paper, we provide partial answers to the above open problems and among other results, we establish the following:

• 1. For every integer $n\ge 2$, if $\mathsf {RC}_{i}$ is true for all integers i with $2\le i\le n$, then $\mathsf {C}_{i}^{-}$ is true for all integers i with $2\le i\le n$.

• 2. If $m,n\ge 2$ are any integers such that for some prime p we have $p\not \mid m$ and $p\mid n$, then in $\mathsf {ZF}$: $\mathsf {RC}_{m}\nrightarrow \mathsf {RC}_{n}$ and $\mathsf {RC}_{m}\nrightarrow \mathsf {C}_{n}^{-}$.

• 3. For $n=2,3$, $\mathsf {RC}_{5}$$+$$\mathsf {C}_{n}^{-}$ implies $\mathsf {C}_{5}^{-}$, and $\mathsf {RC}_{5}$ implies neither $\mathsf {C}_{2}^{-}$ nor $\mathsf {C}_{3}^{-}$ in $\mathsf {ZF}$.

• 4. For every integer $k\ge 2$, $\mathsf {RC}_{2k}$ implies “every infinite linearly orderable family of k-element sets has a partial Kinna–Wagner selection function” and the latter implication is not reversible in $\mathsf {ZF}$ (for any $k\in \omega \backslash \{0,1\}$). In particular, $\mathsf {RC}_{6}$ strictly implies “every infinite linearly orderable family of 3-element sets has a partial choice function”.

• 5. The Chain-AntiChain Principle (“every infinite partially ordered set has either an infinite chain or an infinite anti-chain”) implies neither $\mathsf {RC}_{n}$ nor $\mathsf {C}_{n}^{-}$ in $\mathsf {ZF}$, for every integer $n\ge 2$.