Binding three or four bosons without bound subsystems

We estimate the ratio R=g₃/g₂ of the critical coupling constants g₂ and g₃ that are required to achieve binding of two or three bosons, respectively, with a short-range interaction, and examine how this ratio depends on the shape of the potential. Simple monotonous potentials give R~=0.8. A wide repulsive core pushes this ratio close to R=1. On the other hand, for an attractive well protected by an external repulsive barrier, the ratio approaches the rigorous lower bound R=2/3. We also present results for N=4 bosons, sketch the extension to N>4, and discuss various consequences.