TITLE: Points at integer distance
QUESTION [6 upvotes]: How many points can one can place in $\mathbb{R}^n$, with the requirement that no $n+1$ points lie in the same $\mathbb{R}^{n-1}$-plane, and the euclidean distance between every two points is an integer?

REPLY [2 votes]: With the restriction no three points in a line, no four points on a circle, there is a 7 point configuration on the plane. I came here to ask about the upper bound in 3-space, with the added restriction no 4 points on a plane.
[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)
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