Paley graphs satisfy all first-order adjacency axioms

A graph satisfies Axiom n if, for any sequence of 2n of its points, there is another point adjacent to the first n and not to any of the last n. We show that, for each n, all sufficiently large Paley graphs satisfy Axiom n. From this we conclude at once that several properties of graphs are not first order, including self-complementarity and regularity.