  The Kaluza-Klein compactification in the limit of large number of extra
dimensions is studied. Starting point is the Einstein-Hilbert action plus
cosmological constant in 4+D dimensions. It is shown that in the large D limit
the effective four dimensional cosmological constant is of order 1/D whereas
the size of the extra dimensions remains finite. A 't Hooft like large D
expansion of the effective Lagrangian for the Kaluza-Klein scalar and gauge
fields arising from the dimensional reduction is considered. It is shown that
the propagator of the scalar field associated to the determinant of the metric
of the extra dimensions is strongly suppressed. This is an interesting result
as in standard Kaluza-Klein theory this scalar degree of freedom is responsible
for the constraint on the gauge fields which makes it impossible to recover the
usual Yang-Mills equations. Moreover in the large D limit it turns out that the
ultraviolet divergences due to the interactions between gauge and scalar fields
are softened.
