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

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

Published 2014-08-18Version 1

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 $\Gamma$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $\Gamma$. Further structural information about End $R$ is established including that Aut $\Gamma$ arises in uncountably many ways as a Sch\"{u}tzenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.