Abstract
We investigate the existence of metric spaces which, for any coloring with a fixed number of colors, contain monochromatic isomorphic copies of a fixed starting space K. In the main theorem we construct such a space of size \(2^{\aleph_0}\) for colorings with \(\aleph_0\) colors and any metric space K of size \(\aleph_0\). We also give a slightly weaker theorem for countable ultrametric K where, however, the resulting space has size \(\aleph_1\).
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Acknowledgements
Several people provided valuable input for the paper. The second author would like to thank Peter Komjáth for encouraging discussions and to the members of the Prague Set Theory seminar, in particular David Chodounský and Jan ‘Honza’ Grebík, who patiently listened to my presentations and helpfully pointed out errors. The ideas behind the proof of Theorem 1.2 are entirely due to the first author: the second author has merely deciphered them and filled in the technical details.
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The first author was partially supported by European Research Council grant 338821 and the ISF (Israel Science Foundation) grant 1838(10) for partially supporting this research. The paper has been edited using typing services generously funded by an individual who wishes to remain anonymous. Paper 1123 on author’s list.
The second author was supported by FWF-GAČR LA Grant no. 17-33849L: Filters, ultrafilters and connections with forcing.
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Shelah, S., Verner, J. Ramsey partitions of metric spaces. Acta Math. Hungar. 169, 524–533 (2023). https://doi.org/10.1007/s10474-023-01318-6
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DOI: https://doi.org/10.1007/s10474-023-01318-6