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 E0 in complexity.
This is a post-peer-review, pre-copyedit version of an article published in Archive for Mathematical Logic. The final authenticated version is available online at doi: 10.1007/s00153-018-0651-2
Clemens, John; Coskey, Samuel; and Potter, Stephanie. (2019). "On the Classification of Vertex-Transitive Structures". Archive for Mathematical Logic, 58(5-6), 565-574. https://dx.doi.org/10.1007/s00153-018-0651-2