  Recent observations of high redshift quasar spectra reveal long gaps with
little flux. A small or no detectable flux does not by itself imply the
intergalactic medium (IGM) is neutral. Inferring the average neutral fraction
from the observed absorption requires assumptions about clustering of the IGM,
which the gravitational instability model supplies. Our most stringent
constraint on the neutral fraction at z ~ 6 is derived from the mean Lyman-beta
transmission measured from the z=6.28 SDSS quasar of Becker et al. -- the
neutral hydrogen fraction at mean density has to be larger than 4.7 times
10^{-4}. This is substantially higher than the neutral fraction of ~ 3-5 times
10^{-5} at z = 4.5 - 5.7, suggesting that dramatic changes take place around or
just before z ~ 6, even though current constraints are still consistent with a
fairly ionized IGM at z ~ 6. An interesting alternative method to constrain the
neutral fraction is to consider the probability of having many consecutive
pixels with little flux, which is small unless the neutral fraction is high.
This constraint is slightly weaker than the one obtained from the mean
transmission. We show that while the derived neutral fraction at a given
redshift is sensitive to the power spectrum normalization, the size of the jump
around z ~ 6 is not. We caution that systematic uncertainties include spatial
fluctuations in the ionizing background, and the continuum placement. Tests are
proposed. In particular, the sightline to sightline dispersion in mean
transmission might provide a useful diagnostic. We express the dispersion in
terms of the transmission power spectrum, and develop a method to calculate the
dispersion for spectra that are longer than the typical simulation box.
