  The Complete Calibration of the Colour-Redshift Relation survey (C3R2) is a
spectroscopic effort involving ESO and Keck facilities designed to empirically
calibrate the galaxy colour-redshift relation - P(z|C) to the Euclid depth
(i_AB=24.5) and is intimately linked to upcoming Stage IV dark energy missions
based on weak lensing cosmology. The aim is to build a spectroscopic
calibration sample that is as representative as possible of the galaxies of the
Euclid weak lensing sample. In order to minimise the number of spectroscopic
observations to fill the gaps in current knowledge of the P(z|C),
self-organising map (SOM) representations of the galaxy colour space have been
constructed. Here we present the first results of an ESO@ VLT Large Programme
approved in the context of C3R2, which makes use of the two VLT optical and
near-infrared multi-object spectrographs, FORS2 and KMOS. This paper focuses on
high-quality spectroscopic redshifts of high-z galaxies observed with the KMOS
spectrograph in the H- and K-bands. A total of 424 highly-reliable z are
measured in the 1.3<=z<=2.5 range, with total success rates of 60.7% in the
H-band and 32.8% in the K-band. The newly determined z fill 55% of high and 35%
of lower priority empty SOM grid cells. We measured Halpha fluxes in a 1."2
radius aperture from the spectra of the spectroscopically confirmed galaxies
and converted them into star formation rates. In addition, we performed an SED
fitting analysis on the same sample in order to derive stellar masses, E(B-V),
total magnitudes, and SFRs. We combine the results obtained from the spectra
with those derived via SED fitting, and we show that the spectroscopic failures
come from either weakly star-forming galaxies (at z<1.7, i.e. in the H-band) or
low S/N spectra (in the K-band) of z>2 galaxies.
