Automorphism groups of countable algebraically closed graphs and endomorphisms of the random graph

Igor Dolinka, Robert Gray, Jillian McPhee, James Mitchell, Martyn Quick

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We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph R. As a consequence we show that, for any countable graph Γ, there are uncountably many maximal subgroups of the endomorphism monoid of R isomorphic to the automorphism group of Γ. Further structural information about End R is established including that Aut Γ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.
Original languageEnglish
Pages (from-to)437-462
Number of pages26
JournalMathematical Proceedings of the Cambridge Philosophical Society
Issue number03
Early online date21 Jan 2016
Publication statusPublished - May 2016


  • existentially closed graphs
  • algebraically closed graphs
  • random graph
  • endomorphism monoid
  • countable universal graph
  • countable universal bipartite graph

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