High Quality Content by WIKIPEDIA articles! An Erd?s–Diophantine graph is an object in the mathematical subject of Diophantine geometry consisting of a set of integer points at integer distances in the plane that cannot be extended by any additional points. Equivalently, it can be described as a complete graph with vertices located on the integer square grid Z such that all mutual distances between the vertices are integers, while all other grid points have a non-integer distance to at least one vertex. Erd?s–Diophantine graphs are named after Paul Erd?s and Diophantus of Alexandria. They form a subset of the set of Diophantine figures, which are defined as complete graphs in the Diophantine plane for which the length of all edges are integers. Thus, Erd?s–Diophantine graphs are exactly the Diophantine figures that cannot be extended. The existence of Erd?s–Diophantine graphs follows from the Erd?s–Anning theorem,

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