Abstract
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.
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Acknowledgements
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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Clemens, J., Coskey, S. & Potter, S. On the classification of vertex-transitive structures. Arch. Math. Logic 58, 565–574 (2019). https://doi.org/10.1007/s00153-018-0651-2
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DOI: https://doi.org/10.1007/s00153-018-0651-2