Oh, right... if all pairs of points at distance sqrt(3)d were the same color, then this would also be true of any pair of points at distance d, with your isosceles triangle. So this gives another way to express the challenge of improving the lower bound: is there a unit-distance graph G with a pair of vertices u,v such that, for all 4-colorings of G, u and v are forced to be the same color? (Exercise: if there is, then there is a unit-distance graph G' that is not 4-colorable.) Cris On Apr 3, 2013, at 5:58 PM, Tom Karzes wrote:
The claim I made about all points separated by distance sqrt(3)*d being the same color must be true if in fact all points separated by distance d are different colors, as you said. It is a logical implication of that assumption. But that implication can be used to show that the original assumption cannot hold.