The Volume Fraction of Ionized Intergalactic Gas at Redshift z=6.5
  The observed number density of Lyman-alpha sources implies a minimum volume of the inter-galactic medium that must be ionized, in order to allow the Lyman-alpha photons to escape attenuation. We estimate this volume by assigning to each Lyman-alpha emitter the minimum Stromgren sphere that would allow half its Lyman-alpha photons to escape. This implies a lower limit to ionized gas volume fraction of 20-50% at redshift z=6.5. This is a lower limit in two ways: First, we conservatively assume that the Lyman-alpha sources seen (at a relatively bright flux limit) are the only ones present; and second, we assume the smallest Stromgren sphere volume that will allow the photons to escape. This limit is completely independent of what ionizing photon sources produced the bubbles. Deeper Lyman-alpha surveys are possible with present technology, and can strengthen these limits by detecting a higher density of Lyman-alpha galaxies. 
Sangeeta Malhotra, James Rhoads