First-order axiomatisations of representable relation algebras need\n formulas of unbounded quantifier depth

Using a variation of the rainbow construction and various pebble and\ncolouring games, we prove that RRA, the class of all representable relation\nalgebras, cannot be axiomatised by any first-order relation algebra theory of\nbounded quantifier depth. We also prove that the class At(RRA) of atom\nstructures of representable, atomic relation algebras cannot be defined by any\nset of sentences in the language of RA atom structures that uses only a finite\nnumber of variables.\n