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A Note on Computable Distinguishing Colorings

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Abstract

An \(\alpha\)-coloring \(\xi\) of a structure \(\mathcal{S}\) is distinguishing if there are no nontrivial automorphisms of \(\mathcal{S}\) respecting \(\xi\). In this note we prove several results illustrating that computing the distinguishing number of a structure can be very hard in general. In contrast, we show that every computable Boolean algebra has a \(0^{\prime\prime}\)-computable distinguishing 2-coloring. We also define the notion of a computabile distinguishing \(2\)-coloring of a separable space; we apply the new definition to separable Banach spaces.

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Notes

  1. For a finite metric space we also allow the domain to be an initial segment of \(\omega\). We allow \(d(i,j)=0\) in \(X=(\omega,d)\); a standard trick can be used to remove repetitions.

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Funding

Bazhenov was supported by RFBR, project number 20-31-70006. Greenberg was supported by Grant 17-VUW-090 from the Marsden Fund of New Zealand. Melnikov was supported by the Marsden Fund of New Zealand and the Rutherford Discovery Fellowship. Miller was partially supported by grant no. 581896 from the Simons Foundation and by the City University of New York PSC-CUNY Research Award Program. Ng was partially supported by the MOE grant RG23/19.

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Correspondence to N. Bazhenov, N. Greenberg, A. Melnikov, R. Miller or K. M. Ng.

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(Submitted by I. Sh. Kalimullin)

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Bazhenov, N., Greenberg, N., Melnikov, A. et al. A Note on Computable Distinguishing Colorings. Lobachevskii J Math 42, 693–700 (2021). https://doi.org/10.1134/S1995080221040053

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  • DOI: https://doi.org/10.1134/S1995080221040053

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