The latter ones are called Painlevé's conjecture (no-collisions singularities). Their existence has been conjectured for n > 3 by Painlevé (see Painlevé conjecture). Examples of this behavior for n = 5 have been constructed by Xia and a heuristic model for n = 4 by Gerver. Donald G. Saari has shown that for 4 or fewer bodies, the set of initial data giving rise to singularities has measure zero.