  The discovery of high-amplitude brightness oscillations at the spin frequency
or its first overtone in six neutron stars in low-mass X-ray binaries during
type~1 X-ray bursts provides a powerful new way to constrain the compactness of
these stars, and hence to constrain the equation of state of the dense matter
in all neutron stars. Here we present the results of general relativistic
calculations of the maximum fractional rms amplitudes that can be observed
during bursts. In particular, we determine the dependence of the amplitude on
the compactness of the star, the angular dependence of the emission from the
surface, the rotational velocity at the stellar surface, and whether there are
one or two emitting poles. We show that if two poles are emitting, as is
strongly indicated by independent evidence in 4U 1636-536 and KS 1731-26, the
resulting limits on the compactness of the star can be extremely restrictive.
We also discuss the expected amplitudes of X-ray color oscillations and the
observational signatures necessary to derive convincing constraints on neutron
star compactness from the amplitudes of burst oscillations.
