  A Diophantine figure is a set of points on the integer grid $\mathbb{Z}^{2}$
where all mutual Euclidean distances are integers. We also speak of Diophantine
graphs. In this language a Diophantine figure is a complete Diophantine graph.
Due to a famous theorem of Erd\H{o}s and Anning there are complete Diophantine
graphs which are not contained in larger ones. We call them
Erd\H{o}s-Diophantine graphs. A special class of Diophantine graphs are
Diophantine carpets. These are planar triangulations of a subset of the integer
grid.
  We give an effective construction for Erd\H{o}s-Diophantine graphs and
characterize the chromatic number of Diophantine carpets.
