It's easy to get infinitely many points in the plane with all pairwise distances integers. First of all, if you don't mind them being collinear, just take (n,0). Second, Ptolemy's Theorem implies that if two Pythagorean triangles share a hypotenuse, then their right-angled vertices are a rational distance from one another. This is a recipe for infinitely many points at pairwise rational distances (or any finite number at integer pairwise distances, after scaling), all on a circle. It you want general position, I believe the best known is seven points -- the first such integer heptagon was discovered in 2006 by Tobias Kreisel and Sascha Kurz. Ed Pegg's interactive picture of it is here: http://demonstrations.wolfram.com/LabelingTheIntegerHeptagon/ These are lovely results, but I don't think they have any bearing on the chromatic number of the plane :-/. --Michael On Thu, Apr 4, 2013 at 1:40 PM, Cris Moore <moore@santafe.edu> wrote:
aha! more generally, is it impossible to build a K_5 whose edge lengths are sqrt(integer)s?
Cris
On Apr 3, 2013, at 11:06 PM, Warren D Smith wrote:
Every distance between two points of the eq.tri. lattice is sqrt(integer) and obviously the ratio of two such, cannot be phi, since it must be sqrt(rational) and phi^2=phi+1=irrational.
Cristopher Moore Professor, Santa Fe Institute
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