  We investigate the mechanical instability toward density collapse and the
transition temperature of Bose-Einstein condensation in the weakly coupled
boson-fermion many-body mixture of single-component boson and fermion gases, in
the case of the repulsive boson-boson and attractive boson-fermion
interactions; no fermion-fermion interaction is assumed due to the Pauli
exclusion because of the diluteness of the mixture. The mechanical instability
occurs in the part of bosonic gas in the case where the induced boson-boson
attraction mediated by the fermion polarization overcomes the boson-boson
repulsive interaction at low temperatures. We calculate the onset temperature
of the mechanical instability and the BEC transition temperature for the
interaction strength and the boson-fermion number ratio. We also discuss the
relation between the mechanical instability and the BEC transition using a
diagrammatic method.
