  We investigate clustering properties of dark matter halos and galaxies to
search for optimal statistics and scales where possible departures from general
relativity (GR) could be found. We use large N-body cosmological simulations to
perform measurements based on the two-point correlation function (2PCF) in GR
and in selected modified gravity (MG) structure formation scenarios. As a
test-bed, we employ two popular beyond-GR models: $f(R)$ gravity and the normal
branch of the Dvali-Gabadadze-Porrati (nDGP) braneworld. We study a range of
simulated halo and galaxy populations and reveal a noticeable MG signal in the
monopole and quadrupole moments of the redshift-space 2PCF, and in the
so-called clustering wedges. However, once expressed in terms of the linear
distortion parameter, $\beta$, the statistical significance of these signals
largely diminishes due to a strong degeneracy between MG-enhanced clustering
and modified tracer bias. To circumvent this, we consider statistics less
dependent on the bias: relative clustering ratios. We generalize the monopole
ratio proposed in earlier work to multipole moments and clustering wedges, and
introduce a new estimator of the $\beta$ parameter. The clustering ratios we
extract foster noticeable differences between MG and GR models, reaching a
maximum deviation of 10\% at 2$\sigma$ significance for specific variants of
$f(R)$ and nDGP. We show that such departures could be measured for $\beta$ if
non-linear effects at intermediate scales are correctly modeled. Our study
indicates that the clustering ratios give great promise to search for
signatures of MG in the large-scale structure. We also find that the selection
of an optimal tracer sample depends on a particular statistics and gravity
model to be considered.
