A corrected strategy for proving no finite variable axiomatisation exists for RRA

Abstract

We show that if for all finite $c$ there is a pair of non-isomorphic finite digraphs satisfying some additional conditions, one of which is that they cannot be distinguished in a certain $c$-colour node colouring game, then there can be no axiomatisation of the class of representable relation algebras in any first-order theory of arbitrary quantifier-depth using only finitely many variables. This corrects the proposed strategy of Hirsch and Hodkinson, \emph{Relation algebras by games}, North-Holland (2002), Problem 1. However, even for $c=2$, no pair of non-isomorphic graphs indistinguishable in the game is currently known.