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: