Kepler red-clump stars in the field and in open clusters: constraints on
  core mixing
  Convective mixing in Helium-core-burning (HeCB) stars is one of the
outstanding issues in stellar modelling. The precise asteroseismic measurements
of gravity-modes period spacing ($\Delta\Pi_1$) has opened the door to detailed
studies of the near-core structure of such stars, which had not been possible
before. Here we provide stringent tests of various core-mixing scenarios
against the largely unbiased population of red-clump stars belonging to the old
open clusters monitored by Kepler, and by coupling the updated precise
inference on $\Delta\Pi_1$ in thousands field stars with spectroscopic
constraints. We find that models with moderate overshooting successfully
reproduce the range observed of $\Delta\Pi_1$ in clusters. In particular we
show that there is no evidence for the need to extend the size of the
adiabatically stratified core, at least at the beginning of the HeCB phase.
This conclusion is based primarily on ensemble studies of $\Delta\Pi_1$ as a
function of mass and metallicity. While $\Delta\Pi_1$ shows no appreciable
dependence on the mass, we have found a clear dependence of $\Delta\Pi_1$ on
metallicity, which is also supported by predictions from models.
