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Perfect independent sets with respect to infinitely many relations


We prove a result on perfect cliques with respect to countably many \(G_{\delta }\) relations on a complete metric space. As an application, we show that a Polish group contains a free subgroup generated by a perfect set as long as it contains any uncountable free subgroup. This answers a recent question of Gła̧b and Strobin.


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Author information

Correspondence to Wiesław Kubiś.

Additional information

Research of Martin Doležal was supported by RVO: 67985840. Research of Wiesław Kubiś was supported by GAČR grant P201 14 07880S and RVO: 67985840.

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Doležal, M., Kubiś, W. Perfect independent sets with respect to infinitely many relations. Arch. Math. Logic 55, 847–856 (2016). https://doi.org/10.1007/s00153-016-0498-3

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  • Perfect clique
  • Free subgroup
  • Open relation

Mathematics Subject Classification

  • Primary 03E05
  • 03E15
  • Secondary 54H05
  • 20E05
  • 20F38
  • 20K20