Minimal functions on the random graph

TL;DR

This paper identifies a minimal system of 14 fundamental functions on the random graph, showing how all other non-trivial functions relate to this system through composition and automorphisms, with implications for model theory.

Contribution

It introduces a minimal system of functions on the random graph and applies Ramsey theory to classify functions and reducts, refining existing theorems in model theory.

Findings

A system of 14 functions generates all non-trivial functions on the random graph.

Re-derivation and refinement of the classification of reducts of the random graph.

Proof that all reducts of the random graph are model-complete.

Abstract

We show that there is a system of 14 non-trivial finitary functions on the random graph with the following properties: Any non-trivial function on the random graph generates one of the functions of this system by means of composition with automorphisms and by topological closure, and the system is minimal in the sense that no subset of the system has the same property. The theorem is obtained by proving a Ramsey-type theorem for colorings of tuples in finite powers of the random graph, and by applying this to find regular patterns in the behavior of any function on the random graph. As model-theoretic corollaries of our methods we re-derive a theorem of Simon Thomas classifying the first-order closed reducts of the random graph, and prove some refinements of this theorem; also, we obtain a classification of the minimal reducts closed under primitive positive definitions, and prove that all reducts of the random graph are model-complete.