Very interesting question, whether or not I may have heard it before. And Ed Pegg's graphs are fantastic. I suppose the same question can be asked about a metric surface, especially one with a lot of symmetry like the sphere or one of the tori (here C = complexes, and ζ_n denotes the primitive nth root of unit with smallest positive angle). T_g = C / Z[ζ_4] (the Gaussian torus) and T_e = C / Z[ζ_6] (the Eisenstein torus). (Here Z[α] denotes the subring of the complex numbers generated by the complex number α, i.e. the set of values {P(α)} where P runs through all polynomials of any degree with integer coefficients. The expression C / Z[α] means the quotient of C by the additive group of Z[α], viewed as translations of the complex plane.) The choice of distance to try to repeat is not scale-free, as it is on the plane. Still, the distance of 1 has special significance on T_g and T_e since it is a generator of the additive groups of Z[ζ_4] and Z[ζ_6]. T_e has a *special* relationship with this problem because of all the equilateral triangles among the points of Z[ζ_6]. In any case see the set of vertices of many toral tilings here: [http://www.weddslist.com/groups/genus/1/]. —Dan Allan Wechsler wrote: ----- For a given set of k points in the plane, some number n of the mutual distances are 1. How big can we make n for a given k? A186705 at OEIS given the answers we know, but only up to 14 vertices. Where can I see pictures of example maximal unit distance graphs? The first "surprise" is a(9)=18; I would love to see that one. -----