Implications of Ramsey Choice principles in ZF

Lorenz Halbeisen; Riccardo Plati; Saharon Shelah

doi:10.1002/malq.202300024

Implications of Ramsey Choice principles in ZF

Lorenz Halbeisen^*, Riccardo Plati, Saharon Shelah

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The Ramsey Choice principle for families of (Formula presented.) -element sets, denoted (Formula presented.), states that every infinite set (Formula presented.) has an infinite subset (Formula presented.) with a choice function on (Formula presented.). We investigate for which positive integers (Formula presented.) and (Formula presented.) the implication (Formula presented.) is provable in (Formula presented.). It will turn out that beside the trivial implications (Formula presented.), under the assumption that every odd integer (Formula presented.) is the sum of three primes (known as ternary Goldbach conjecture), the only non-trivial implication which is provable in (Formula presented.) is (Formula presented.).

Original language	English
Pages (from-to)	255-261
Number of pages	7
Journal	Mathematical Logic Quarterly
Volume	70
Issue number	2
DOIs	https://doi.org/10.1002/malq.202300024
State	Published - May 2024

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© 2024 The Author(s). Mathematical Logic Quarterly published by Wiley-VCH GmbH.

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abstract = "The Ramsey Choice principle for families of (Formula presented.) -element sets, denoted (Formula presented.), states that every infinite set (Formula presented.) has an infinite subset (Formula presented.) with a choice function on (Formula presented.). We investigate for which positive integers (Formula presented.) and (Formula presented.) the implication (Formula presented.) is provable in (Formula presented.). It will turn out that beside the trivial implications (Formula presented.), under the assumption that every odd integer (Formula presented.) is the sum of three primes (known as ternary Goldbach conjecture), the only non-trivial implication which is provable in (Formula presented.) is (Formula presented.).",

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Implications of Ramsey Choice principles in ZF

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