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We show that if for all finite $c$ there is a pair of non-isomorphic finitedigraphs satisfying some additional conditions, one of which is that theycannot be distinguished in a certain $c$-colour node colouring game, then therecan be no axiomatisation of the class of representable relation algebras in anyfirst-order theory of arbitrary quantifier-depth using only finitely manyvariables. 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 gameis currently known.

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