  We investigate the role of electron-electron and electron-phonon interactions
in strongly correlated systems by performing unbiased quantum Monte Carlo
simulations in the square lattice Hubbard-Holstein model at half-filling. We
study the competition and interplay between antiferromagnetism (AFM) and
charge-density wave (CDW), establishing its very rich phase diagram. In the
region between AFM and CDW phases, we have found an enhancement of
superconducting pairing correlations, favoring (nonlocal) $s$-wave pairs. Our
study sheds light over past inconsistencies in the literature, in particular
the emergence of CDW in the pure Holstein model case.
