  We present the results of three separate searches for HI 21-cm absorption in
a total of twelve damped Lyman-alpha absorption systems (DLAs) and sub-DLAs
over the redshift range z = 0.86-3.37. We find no absorption in the five
systems for which we obtain reasonable sensitivities and add the results to
those of other recent surveys in order to investigate factors which could have
an effect on the detection rate: We provide evidence that the mix of spin
temperature/covering factor ratios seen at low redshift may also exist at high
redshift, with a correlation between the 21-cm line strength and the total
neutral hydrogen column density, indicating a roughly constant spin
temperature/covering factor ratio for all of the DLAs searched. Also, by
considering the geometry of a flat expanding Universe together with the
projected sizes of the background radio emission regions, we find, for the
detections, that the 21-cm line strength is correlated with the size of the
absorber. For the non-detections it is apparent that larger absorbers (covering
factors) are required in order to exhibit 21-cm absorption, particularly if
these DLAs do not arise in spiral galaxies. We also suggest that the recent z =
2.3 detection towards TXS 0311+430 arises in a spiral galaxy, but on the basis
of a large absorption cross-section and high metallicity, rather than a low
spin temperature
