Stress, rigidity and rhombic reductions for an Erdos minimum-diameter problem (Problem #99)

For n ≥ 2, let Dn be the least possible diameter of an n-point subset of the Euclideanplane whose minimum pairwise distance is 1. Erd˝os asked whether every configuration attainingDn contains a unit equilateral triangle for all sufficiently large n. We give structural reductionstoward this open problem. First, we formulate diameter minimization as a nonlinear programand prove that every minimizer satisfies the Mangasarian–Fromovitz constraint qualification.Hence it admits a nonnegative equilibrium stress balancing unit contacts against diametral pairs.A perturbation argument yields an adapted stress supported on contacts that are frozen byevery infinitesimal contact flex fixing all diametral pairs pointwise. If S is the vertex set ofits positive-contact support, then |E1| ≥ 2|S| − O(√n). Thus a triangle-free counterexampleeither has a large unstressed sector or has 2n − O(√n) unit contacts. In the latter case itsplanar contact graph has n − O(√n) unit-rhombus faces, with total angular excess over 60degrees equal to O(√n). We also exclude sufficiently large rigid counterexamples contained ina two-direction unit-rhombic lattice, and unconditionally exclude convex unit-rhombus tilings.Finally, train-track coordinates reduce the remaining obstruction to a boundary-pinning problemof order √n.