Infinite Paley graphs

TL;DR

This paper constructs infinite Paley graphs using uncountably many infinite locally finite fields, demonstrating their isomorphism to the universal graph and exploring their automorphism groups and model-theoretic connections.

Contribution

It introduces a novel construction of infinite Paley graphs based on uncountably many infinite fields and analyzes their properties and symmetries.

Findings

All constructed graphs are isomorphic to the universal graph

Automorphism groups are characterized

Connections with model theory are established

Abstract

Infinite analogues of the Paley graphs are constructed, based on uncountably many infinite but locally finite fields. Weil's estimate for character sums shows that they are all isomorphic to the random or universal graph of Erd\H os, R\'enyi and Rado. Automorphism groups and connections with model theory are considered.