The conjugacy problem for automorphism groups of homogeneous digraphs

Samuel Coskey, Paul Ellis


We decide the Borel complexity of the conjugacy problem for automorphism groups of countable homogeneous digraphs. Many of the homogeneous digraphs, as well as several other homogeneous structures, have already been addressed in previous articles. In this article we complete the program, and establish a dichotomy theorem that this complexity is either the minimum or the maximum among relations which are classifiable by countable structures. We also discuss the possibility of extending our results beyond graphs to more general classes of countable homogeneous structures.


conjugacy, homogeneous structure, Borel complexity

Full Text:



[BM13] Dogan Bilge and Julien Melleray. Elements of finite order in automorphism groups of homogeneous structures. Contrib. Discrete Math., 8(2):88–119, 2013.

[CE15] Samuel Coskey and Paul Ellis. The conjugacy problem for automorphism groups of countable homogeneous structures. Preprint, 2015, arXiv:1406.6411.

[CES11] Samuel Coskey, Paul Ellis, and Scott Schneider. The conjugacy problem for the automorphism group of the random graph. Arch. Math. Logic, 50(1-2):215–221, 2011.

[Che87] Gregory L. Cherlin. Homogeneous directed graphs. The imprimitive case. In Logic colloquium ’85 (Orsay, 1985), volume 122 of Stud. Logic Found. Math., pages 67–88. North-Holland, Amsterdam, 1987.

[Che98] Gregory L. Cherlin. The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments. Mem. Amer. Math. Soc., 131(621):xiv+161, 1998.

[FS89] Harvey Friedman and Lee Stanley. A Borel reducibility theory for classes of countable structures. J. Symbolic Logic, 54(3):894–914, 1989.

[Gao09] Su Gao. Invariant descriptive set theory, volume 293 of Pure and Applied Mathematics (Boca Raton). CRC Press, Boca Raton, FL, 2009.

[Mac11] Dugald Macpherson. A survey of homogeneous structures. Discrete Math., 311(15):1599–1634, 2011.

[Sch79] James H. Schmerl. Countable homogeneous partially ordered sets. Algebra Universalis, 9(3):317–321, 1979.


Contributions to Discrete Mathematics. ISSN: 1715-0868