The first (nearly) model-independent constraint on the neutral hydrogen fraction at

Cosmic reionization is expected to be complex, extended and very inhomogeneous. Existing constraints atz~ 6 on the volume-averaged neutral hydrogen fraction,, are highly model-dependent and controversial. Constraints atz< 6, suggesting that the Universe is highly ionized, are also model-dependent, but more fundamentally are invalid in the context of inhomogeneous reionization. As such, it has recently been pointed out that there is no conclusive evidence that reionization has completed byz~ 5-6, a fact that has important ramifications on the interpretation of high-redshift observations and theoretical models. We present the first direct upper limits onatz~ 5-6 using the simple and robust statistic of the covering fraction of dark pixels in the Lyα/β forests of high-redshift quasars. With a sample of 13 Keck Echellette Spectrograph and Imager spectra we constrainat 5.0 ≲z≲ 5.5, rising toatz~ 6.1. We also find tentative evidence for a break in the redshift evolution of the dark covering fraction atz~ 5.5. A subsample of two deep spectra provides a more stringent constraint of, when combined with conservative estimates of cosmic variance. This upper limit is comparable to existing results atz~ 6 but is more robust. The results presented here do not rely on assumptions about quasar continua, intergalactic medium density, the morphology of Hii regions or ionizing background fields, and thus are a good starting point for future interpretation of high-redshift observations.