Euclid preparation XX. The Complete Calibration of the Color-Redshift Relation survey: LBT observations and data release

The Complete Calibration of the Color-Redshift Relation survey (C3R2) is a spectroscopic program 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 that is 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-band 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