  The Complete Calibration of the Color-Redshift Relation survey (C3R2) is a
spectroscopic programme designed to empirically calibrate the galaxy
color-redshift relation to the Euclid depth (I_E=24.5), a key ingredient for
the success of Stage IV dark energy projects based on weak lensing cosmology. A
spectroscopic calibration sample as representative as possible of the galaxies
in the Euclid weak lensing sample is being collected, selecting galaxies from a
self-organizing map (SOM) representation of the galaxy color space. Here, we
present the results of a near-infrared H- and K-bands spectroscopic campaign
carried out using the LUCI instruments at the LBT. For a total of 251 galaxies,
we present new highly-reliable redshifts in the 1.3<= z <=1.7 and 2<= z<=2.7
ranges. The newly-determined redshifts populate 49 SOM cells which previously
contained no spectroscopic measurements and almost double the occupation
numbers of an additional 153 SOM cells. A final optical ground-based
observational effort is needed to calibrate the missing cells in particular in
the redshift range 1.7<= z<=2.7 that lack spectroscopic calibration. In the
end, Euclid itself will deliver telluric-free NIR spectra that can complete the
calibration.
