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We consider a system of $N$ pairwise interacting particles described by theHamiltonian $H$, where $\sigma_{ess} (H) = [0,\infty)$ and none of the particlepairs has a zero energy resonance. The pair potentials are allowed to take bothsigns and obey certain restrictions regarding the fall off. It is proved thatif $N \geq 4$ and none of the Hamiltonians corresponding to the subsystemscontaining $N-2$ or less particles has an eigenvalue equal to zero then $H$ hasa finite number of negative energy bound states. This result provides apositive proof to a long--standing conjecture of Amado and Greenwood statingthat four bosons with an empty negative continuous spectrum have at most afinite number of negative energy bound states. Additionally, we give a shortproof to the theorem of Vugal'ter and Zhislin on the finiteness of the discretespectrum and pose a conjecture regarding the existence of the "true" four--bodyEfimov effect.

Why there is no Efimov effect for four bosons and related results on the finiteness of the discrete spectrum