Erdős–Anning theorem

Remarks

Suppose that $S$ contains three points $A, B, C$ that are not collinear. For any other point $X$ of $S$, the triangle inequality gives $\abs{\,\norm{X - A} - \norm{X - B}\,} \le \norm{A - B}$, and this difference is an integer. So $X$ lies on one of finitely many hyperbolas with foci $A$ and $B$, counting degenerate ones such as the perpendicular bisector and parts of the line $AB$. In the same way $X$ lies on one of finitely many hyperbolas with foci $A$ and $C$, and two such curves with different foci meet in at most $4$ points. So $S$ is finite, a contradiction. The hypothesis that $S$ is infinite is needed: the vertices of a $3$-$4$-$5$ right triangle are three non-collinear points with integer distances. Integer distances cannot be weakened to rational ones: there are infinite non-collinear sets with all distances rational, for example dense subsets of a circle.

Going deeper

For a student

Norman Anning and Paul Erdős proved the theorem in 1945. The proof is a short argument with hyperbolas, using the definition of a hyperbola as the set of points with a fixed difference of distances to two foci. In contrast, for every $n$ there are $n$ points in the plane, no three on a line, with all distances integers; one can take suitable points on a circle.