  The fiducial cosmological analyses of imaging galaxy surveys like the Dark
Energy Survey (DES) typically probe the Universe at redshifts $z < 1$. This is
mainly because of the limited depth of these surveys, and also because such
analyses rely heavily on galaxy lensing, which is more efficient at low
redshifts. In this work we present the selection and characterization of
high-redshift galaxy samples using DES Year 3 data, and the analysis of their
galaxy clustering measurements. In particular, we use galaxies that are fainter
than those used in the previous DES Year 3 analyses and a Bayesian redshift
scheme to define three tomographic bins with mean redshifts around $z \sim
0.9$, $1.2$ and $1.5$, which significantly extend the redshift coverage of the
fiducial DES Year 3 analysis. These samples contain a total of about 9 million
galaxies, and their galaxy density is more than 2 times higher than those in
the DES Year 3 fiducial case. We characterize the redshift uncertainties of the
samples, including the usage of various spectroscopic and high-quality redshift
samples, and we develop a machine-learning method to correct for correlations
between galaxy density and survey observing conditions. The analysis of galaxy
clustering measurements, with a total signal-to-noise $S/N \sim 70$ after scale
cuts, yields robust cosmological constraints on a combination of the fraction
of matter in the Universe $\Omega_m$ and the Hubble parameter $h$, $\Omega_m h
= 0.195^{+0.023}_{-0.018}$, and 2-3% measurements of the amplitude of the
galaxy clustering signals, probing galaxy bias and the amplitude of matter
fluctuations, $b \sigma_8$. A companion paper $\textit{(in preparation)}$ will
present the cross-correlations of these high-$z$ samples with CMB lensing from
Planck and SPT, and the cosmological analysis of those measurements in
combination with the galaxy clustering presented in this work.
