Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$.
Say that $S$ is an integer-distance set if every pair of points in $S$ is separated
by an integer Euclidean distance.
What are examples of maximal integer distance sets?
(Maximal: no point can be added while retaining the integer-distance property
between all pairs.)
Of course the lattice points along any one line parallel to a coordinate axis in
$\mathbb{R}^d$ constitute a countably infinite integer-distance set.
What is an example of an infinite integer-distance set of noncollinear points?
I know that Euler established that every circle contains a dense rational-distance set.
Scaling any one circle by a large common denominator provides a rich, but finite integer-distance set. For example, these five points on a circle are all separated by integer
distances:
$$
(1221025, 0), (781456, 586092),
(439569, 586092),
(270400, 507000),
(180625, 433500)
$$
I'm sure this is all known… Thanks for enlightening me!
(This is tangentially related to my earlier question,
"Rational points on a sphere in $\mathbb{R}^d$.")
Update1.
It turns out that determining the integer-distance sets is
fundamentally open.
What is known is nicely summarized by Robert Israel and "Daniel m3."
In particular,
via the Kreisel & Kurz paper,
it is unknown (or was unknown in 2008)
whether or not there exists an 8-point
integer-distance set in $\mathbb{R}^2$,
with no three of the points collinear and no four cocircular.
Update2.
Also open is a related problem identified by Nathan Dean:
How many non-cocircular integer-distance points can be found on a parabola,
a scaling of $y = x^2$?
Nathan proved there are infinitely many sets of three such points;
Garikai Cambell proved there are infinitely many sets of four such points.
But the existence of five such points seems open.
I just learned the parabola problem from this MSE question.
Update3 (21 Jul 2013).
I ran across this just-published paper, which explores the in-some-sense obverse of the
question I asked: What are the largest point sets in $\mathbb{R}^d$ that
avoid points an integral distance apart.
Kurz, Sascha, and Valery Mishkin. "Open Sets Avoiding Integral Distances." Discrete & Computational Geometry (2013): 1-25. (Springer link)
Update4 (29 Nov 2014). There is a nice article at Dick Lipton's blog
on Ulam's 70-year-old un-resolved conjecture:
If $S$ is an rational-distance set, then it is not dense in the plane.
And that article cites the Kurz-Mishkin paper above.
Best Answer
See e.g. http://www.ics.uci.edu/~eppstein/junkyard/integer-distances.html for a proof (originally due to Erdos) that there is no infinite non-collinear integer-distance set in the plane.