El problema de Erdös y Ulam

Abstract

Paul Erdös, in the first half of last century, asked about finite integer sets in the plane in general position. This means, finite many points sets of the plane, with no 3 points on a line, nor 4 on a circle, with integer distances between any 2 points of the set. One can easily find a 3 points set where all distances between those points are integers. However, the problem becomes harder to solve when we try to find greater finite integer sets in general position. We will see some constructions of these sets of $n$ points, being $n < 8$. Nowadays, we know some 7 points integer sets in general position, but we have not found an example of an 8 points set that satisfies the conditions. Stanislaw Ulam studied the infinite version of the problem and conjectured that there is no everywhere dense rational set in the plane. Till today, this problem has been addressed by studying the set of rational points on algebraic curves, achieving the main results from the Mordell’s Theorem and the Faltings’ Theorem.