  We prove that the Maxwell-Klein-Gordon equations on $\R^{1+4}$ relative to
the Coulomb gauge are locally well-posed for initial data in $H^{1+\epsilon}$
for all $\epsilon > 0$. This builds on previous work by Klainerman and Machedon
who proved the corresponding result for a model problem derived from the
Maxwell-Klein-Gordon system by ignoring the elliptic features of the system, as
well as cubic terms.
