There are integral heptagons, no three points on a line, no four on a circle

TL;DR

This paper presents two specific configurations of seven points in the plane that are pairwise integral distances, with no three collinear points and no four cocircular points, answering a longstanding question of Erdős.

Contribution

The authors construct explicit examples of seven-point configurations with integral distances and no three collinear or four cocircular points, solving a famous open problem.

Findings

Provided explicit configurations of seven points with integral pairwise distances.

Confirmed the existence of such configurations answering Erdős's question.

Demonstrated the feasibility of arrangements with these constraints.

Abstract

We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erd\H{o}s.