Given n points {x_1, ..., x_n} on the surface of a 2-sphere S^2, let's call any 3 of them "empty" if they lie on the curve of some circle C ⊂ S^2 that's smaller than a great circle and there is no other point x_j on or inside C. Given n, what is the maximum number of empty triples possible? I can more or less prove that for n = 4, 6, 12 the answer comes from the vertices of a regular tetrahedron, octahedron, icosahedron, giving 4, 8, 20 triples, respectively. Likewise for higher dimensions. Next case: Given n points {x_1, ..., x_n} on the surface of a 3-sphere S^3, call any 4 of them "empty" if they lie on the surface of some 2-sphere S ⊂ S^3 that's smaller than a great sphere and there is no other point x_j on or inside S. I'd certainly guess that the maximum number of empty quadruples for the values n = 5, 8, 10, 120 are 5, 16, 30, 600, respectively. The arrangements would be the vertices of the following: the 5-cell (4-simplex), the 16-cell (a.k.a. 4-orthoplex), the 5-cell and the antipodal 5-cell, the 600-cell, respectively. (Small perturbations of all these arrangements give the same numbers as well.) But what about the in-between values, and asymptotics? —Dan