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We wish to conclude finally by commenting on three- body losses and mechanical instabilities, that could pre- vent the experimental observation of the many-body physics described above. The choice of restricting our analysis to mixtures with a majority of fermions was precisely aimed at reducing the importance of these nui- sances. Three-body losses are dominated by processes involving two bosons and one fermion, with a rate propor- tional to n 2 $_{B}$n$_{F}$ [22]. They can thus be controlled by keep- ing n$_{B}$ sufficiently small with respect to n$_{F}$ . A predomi- nance of fermions should also reduce the tendency to col- lapse, because of the stabilizing effect of Fermi pressure. As a matter of fact, we have verified that the compress- ibility matrix ∂µ$_{s}$/∂n$_{s}$ ′ remained positive definite for all couplings, temperatures, and densities n$_{F}$ ≥ n$_{B}$ consid- ered in our calculations. Apparently, pairing correlations act to protect the system from the mean field instabili- ties dominating the phase diagram of non-resonant Bose- Fermi mixtures. Resonant Bose-Fermi mixtures with a majority of fermions appear thus promising systems for the observation of a rich many-body physics. We thank F. Palestini, A. Perali, and G.C. Strinati for a careful reading of the manuscript. Partial support by the Italian MIUR under Contract Cofin-2007 “Ultracold atoms and novel quantum phases” is acknowledged.