Abstract
The first-order part of Ramsey’s theorem for pairs with an arbitrary number of colors is known to be precisely BΣ 03 . We compare this to the known division of Ramsey’s theorem for pairs into the weaker principles, EM (the Erdős–Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond IΣ 02 is entirely due to the arbitrary color analog of ADS.
Specifically, we show that ADS for an arbitrary number of colors implies BΣ 03 while EM for an arbitrary number of colors is Π 11 -conservative over IΣ 02 and it does not imply IΣ 02 .
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Acknowledgements
The authors thank Ludovic Patey for helpful discussions and comments. The proof of Theorem 6.5 is essentially due to him. The work of the first author is partially supported by NSF grant DMS-1600263. The work of the second author is partially supported by JSPS KAKENHI (grant numbers 19K03601 and 21KK0045).
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Towsner, H., Yokoyama, K. Erdős–Moser and IΣ2. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2643-8
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DOI: https://doi.org/10.1007/s11856-024-2643-8