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

TL;DR

This paper explores special integer-coordinate point sets with all mutual distances integral, introduces Erdős-Diophantine graphs that are maximal, and characterizes the coloring properties of Diophantine carpets, advancing understanding of these geometric structures.

Contribution

It provides an effective construction method for Erdős-Diophantine graphs and characterizes the chromatic number of Diophantine carpets, revealing new structural insights.

Findings

Constructed Erdős-Diophantine graphs explicitly

Characterized chromatic number of Diophantine carpets

Identified maximality conditions for Diophantine graphs

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.