  We show that in dimensions $n \geq 6$ that one has global regularity for the
Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical
Sobolev norm $\dot H^{n/2-1} \times \dot H^{n/2-2}$ of the initial data is
sufficiently small. These results are analogous to those recently obtained for
the high-dimensional wave map equation but unlike the wave map equation, the
Coulomb gauge non-linearity cannot be iterated away directly. We shall use a
different approach, proving Strichartz estimates for the covariant wave
equation. This in turn will be achieved by use of Littlewood-Paley multipliers,
and a global parametrix for the covariant wave equation constructed using a
truncated, microlocalized Cronstrom gauge.
