  Data from four Fermi-detected gamma-ray bursts (GRBs) is used to set limits
on spectral dispersion of electromagnetic radiation across the universe. The
analysis focuses on photons recorded above 1 GeV for Fermi detected GRB
080916C, GRB 090510A, GRB 090902B, and GRB 090926A because these high-energy
photons yield the tightest bounds on light dispersion. It is shown that
significant photon bunches in GRB 090510A, possibly classic GRB pulses, are
remarkably brief, an order of magnitude shorter in duration than any previously
claimed temporal feature in this energy range. Although conceivably a $>3
\sigma$ fluctuation, when taken at face value, these pulses lead to an order of
magnitude tightening of prior limits on photon dispersion. Bound of $\Delta c /
c < 6.94$ x $10^{-21}$ is thus obtained. Given generic dispersion relations
where the time delay is proportional to the photon energy to the first or
second power, the most stringent limits on the dispersion strengths were $k_1
<$ 1.61 x $10^{-5}$ sec Gpc$^{-1}$ GeV$^{-1}$ and $k_2 <$ 3.57 x $10^{-7}$ sec
Gpc$^{-1}$ GeV$^{-2}$ respectively. Such limits constrain dispersive effects
created, for example, by the spacetime foam of quantum gravity. In the context
of quantum gravity, our bounds set $M_1 c^2$ greater than 525 times the Planck
mass, suggesting that spacetime is smooth at energies near and slightly above
the Planck mass.
