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With the restriction no three points in a line, no four points on a circle, there is a 7 point configuration of points on the plane such that all pairs of points are at integer distances. [1]

For 3-space, add the restriction of no 4 points in a plane. One way to get 5 points at integral distances is to first find a set of 5 Soddy spheres that have integral curvature (curvature=1/radius). For example: $\{1, 3, 4, 9, 16\}$. This set satisfies $3(1^2+3^2+4^2+9^2+16^2)=(1+3+4+9+16)^2$, so it's a set of Soddy spheres. That corresponds to five touching spheres with radii $\{144, 48, 36, 16, 9\}$. The ten distances between points are all integer values.

Is there a set of 5 points with smaller distances?

Are there solutions for 6 points or more?

[1] Tobias Kreisel and Sascha Kurz, There are integral heptagons, no three points on a line, no four on a circle, Discrete Comput. Geom. 39 (2008), no. 4, 786–790, MR2413160 (2009d:52021) labeled diagram

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Note that you can use "\{" and "\}" for $\{$ and $\}$, respectively, rather than using { and } (which do not look as nice). –  JavaMan Jan 24 '13 at 6:15
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