A note on Erd\H{o}s-Diophantine graphs and Diophantine carpets

Abstract

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.