Abstract
We consider finite colourings of finite products \(X_1\times X_2\times \cdot \cdot \cdot \times X_n\) of infinite sets and determine what is the minimal number of colours a subproduct \(Y_1\times Y_2\times \cdot \cdot \cdot \times Y_n\) of infinite subsets could achieve.
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Notes
Symmetric and constantly equal to 0 on the diagonal
It is worth mentioning that such a triple A, B, C can be guaranteed not just with \(C = X_n\) but also for any \(C\in \{X_m,\ldots ,X_{n-1}\}\) since otherwise we end up with Case 1 after reindexing the sets. Then the second to last sentence of the Conclusion ensures that the choice \(B = X_n\) is compatible with every \(C \in \{X_m, \ldots ,X_{n-1}\}\).
References
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Acknowledgements
The research on this paper is partially supported by grants from NSERC(455916) and CNRS(UMR7586).
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Todorcevic, S. Colouring finite products. Period Math Hung 84, 31–36 (2022). https://doi.org/10.1007/s10998-021-00389-8
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DOI: https://doi.org/10.1007/s10998-021-00389-8