I have no confidence at all that every squarefree graph is realizable. In fact, I can actually disprove this, except my disproof currently cheats slightly. Namely, the two papers I cited before show that maximal squarefree graph has order V^(3/2) edges, whereas unit-distance graph has O(V^(4/3)) edges, hence not every squarefree graph is realizable as a unit distance graph, QED. The reasons this "cheated" is I never verified the Szekely paper's V^(4/3) bound works on spheres. But aside from that, this seems solid. So anyhow, the squarefree graph thing merely yields an upper bound on f(N). But you can for any particular N work out f(N) by machine by considering every possible N-vertex graph and deciding if it is realizable. It ain't an easy job, but it is doable in principle. In fact you can probably get up to maybe N=10 just manually. We don't need no stinking computers, real men cut triangular thingummies with their bowie knives and move them around on spheres to solve such things. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)