All reducts of the random graph are model-complete

Manuel Bodirsky, Michael Pinsker

Published 2009-03-14, updated 2010-04-12Version 3

We study locally closed transformation monoids which contain the automorphism group of the random graph. We show that such a transformation monoid is locally generated by the permutations in the monoid, or contains a constant operation, or contains an operation that maps the random graph injectively to an induced subgraph which is a clique or an independent set. As a corollary, our techniques yield a new proof of Simon Thomas' classification of the five closed supergroups of the automorphism group of the random graph; our proof uses different Ramsey-theoretic tools than the one given by Thomas, and is perhaps more straightforward. Since the monoids under consideration are endomorphism monoids of relational structures definable in the random graph, we are able to draw several model-theoretic corollaries: One consequence of our result is that all structures with a first-order definition in the random graph are model-complete. Moreover, we obtain a classification of these structures up to existential interdefinability.