Implications of Ramsey Choice Principles in ZF

Lorenz Halbeisen; Riccardo Plati; Saharon Shelah

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arXiv · 2306.00743

Implications of Ramsey Choice Principles in ZF

Lorenz Halbeisen·Riccardo Plati·Saharon Shelah

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

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BibTeX RIS

Lorenz Halbeisen, Riccardo Plati, Saharon Shelah. 2023-06-01. Implications of Ramsey Choice Principles in ZF. https://arxiv.org/abs/2306.00743

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