Thoughts on a conjecture of Erdős. Issue 552 (16th October 2017) - Dolan, Stan
Abstract : If two squares with no interior point in common are drawn inside a unit square then prove that the sum of their side-lengths is at most 1. This problem was posed in the 1930s by Paul Erdős [1]. It is the simplest case of a still unsolved conjecture. If k 2 + 1 squares with no interior point in common are drawn inside a unit square then the maximum possible sum of their side-lengths is k [2]. We shall use the notation S(n) to denote the maximum possible sum of the side-lengths for n squares drawn with no interior point in common inside a unit square. The main aim of this article will be to develop an approach to the study of the function S which will give surprisingly simple proofs of a number of known results. This approach will then be used to prove a new result about the asymptotic behaviour of S .