# On the classification of vertex-transitive structures

This post is a link to https://arxiv.org/abs/1707.02383

With John Clemens and Stephanie Potter. *Archive for mathematical logic*, 2019. (vertex-transitive structures in journal)

*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.