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Abstract: We show that there is a system of 14 non-trivial finitary functions on therandom graph with the following properties: Any non-trivial function on therandom graph generates one of the functions of this system by means ofcomposition with automorphisms and by topological closure, and the system isminimal in the sense that no subset of the system has the same property. Thetheorem is obtained by proving a Ramsey-type theorem for colorings of tuples infinite powers of the random graph, and by applying this to find regularpatterns in the behavior of any function on the random graph. Asmodel-theoretic corollaries of our methods we re-derive a theorem of SimonThomas classifying the first-order closed reducts of the random graph, andprove some refinements of this theorem; also, we obtain a classification of theminimal reducts closed under primitive positive definitions, and prove that allreducts of the random graph are model-complete.

Author: Manuel Bodirsky, Michael Pinsker



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