  This is the third in a series of papers that present observations and results
for a sample of 76 ultra-steep-spectrum radio sources designed to find galaxies
at high redshift. Here we present multi-frequency radio observations, from the
Australia Telescope Compact Array, for a subset of 37 galaxies from the sample.
Matched resolution observations at 2.3, 4.8 and 6.2GHz are presented for all
galaxies, with the z<2 galaxies additionally observed at 8.6 and 18GHz. New
angular size constraints are reported for 19 sources based on high resolution
4.8 and 6.2GHz observations. Functional forms for the rest-frame spectral
energy distributions are derived: 89% of the sample is well characterised by a
single power law, whilst the remaining 11% show some flattening toward higher
frequencies: not one source shows any evidence for high frequency steepening.
We discuss the implications of this result in light of the empirical
correlation between redshift and spectral index seen in flux limited samples of
radio galaxies. Finally, a new physical mechanism to explain the redshift --
spectral index correlation is posited: extremely steep spectrum radio galaxies
in the local universe usually reside at the centres of rich galaxy clusters. We
argue that if a higher fraction of radio galaxies, as a function of redshift,
are located in environments with densities similar to nearby rich clusters,
then this could be a natural interpretation for the correlation. We briefly
outline our plans to pursue this line of investigation.
