On the classification of vertex-transitive structures

  • John Clemens
  • Samuel Coskey
  • Stephanie Potter


We consider the classification problem for several classes of countable structures which are “vertex-transitive”, meaning that the automorphism group acts transitively on the elements. (This is sometimes called homogeneous.) We show that the classification of countable vertex-transitive digraphs and partial orders are Borel complete. We identify the complexity of the classification of countable vertex-transitive linear orders. Finally we show that the classification of vertex-transitive countable tournaments is properly above \(E_0\) in complexity.


Borel complexity theory Graphs Linear orders Tournaments 

Mathematics Subject Classification

03E15 05C63 05C20 


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This work represents a portion of the third author’s master’s thesis [7]. The thesis was completed at Boise State University under the supervision of the second author, with significant input from the first author.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Boise State UniversityBoiseUSA

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