Binding three or four bosons without bound subsystems

We estimate the ratio ${R=g}_3{/g}_2$ of the critical coupling constants $g_2$ and $g_3$ 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 $R\simeq \mathrm{0.8.}$ A wide repulsive core pushes this ratio close to $R=1.\mathrm{}$ On the other hand, for an attractive well protected by an external repulsive barrier, the ratio approaches the rigorous lower bound $R=2/3.\mathrm{}$ We also present results for $N=4\mathrm{}$ bosons, sketch the extension to $N>\mathrm{}4,$ and discuss various consequences.