The Efimov effect was originally formulated for three particles. The underlying principle of model independence is extended in this article in several directions. We present our definitions of the concepts of universality and scale independence. In this context we review briefly the scaling relations established for two- and three-body structures of nuclear halos. We emphasize the difference between the two extremes of weak binding named Efimov and Brunnian states. They arise respectively for two-body interactions at threshold of binding either two or N particles. We restrict the Hilbert space to include no more than two-body correlations, and discuss the derived excited N-body Efimov states both for zero- and finite-range two-body interactions. Then we investigate the relation between radius and binding energy extremely close to threshold of binding the Brunnian N-body system. Radii of both ground and first excited states for N = 4, 5, 6 remain finite as the binding energy vanishes, and the distances between pairs of particles are substantially larger than the range of the two-body potential. The radii decrease with N and increase with excitation energy. The computed radii are larger for the complete than for the restricted Hilbert space. Model independence at the Brunnian threshold is strongly indicated.
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