  (Abridged). We introduce the Aarhus Red Giants Challenge, a series of
detailed comparisons between widely used stellar evolution and oscillation
codes aiming at establishing the minimum level of uncertainties in properties
of red giants arising solely from numerical implementations. Using 9
state-of-the-art stellar evolution codes, we defined a set of input physics and
physical constants for our calculations and calibrated the convective
efficiency to a specific point on the main sequence. We produced evolutionary
tracks and stellar structure models at fixed radius along the red-giant branch
for masses of 1.0 M$_\odot$, 1.5 M$_\odot$, 2.0 M$_\odot$, and 2.5 M$_\odot$,
and compared the predicted stellar properties. Once models have been calibrated
on the main sequence we find a residual spread in the predicted effective
temperatures across all codes of ~20 K at solar radius and ~30-40 K in the RGB
regardless of the considered stellar mass. The predicted ages show variations
of 2-5% (increasing with stellar mass) which we track down to differences in
the numerical implementation of energy generation. The luminosity of the
RGB-bump shows a spread of about 10% for the considered codes, which translates
into magnitude differences of ~0.1 mag in the optical V-band. We also compare
the predicted [C/N] abundance ratio and found a spread of 0.1 dex or more for
all considered masses. Our comparisons show that differences at the level of a
few percent still remain in evolutionary calculations of red giants branch
stars despite the use of the same input physics. These are mostly due to
differences in the energy generation routines and interpolation across
opacities, and call for further investigations on these matters in the context
of using properties of red giants as benchmarks for astrophysical studies.
