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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,