Fifty years of the Erdős similarity conjecture

The Erd & odblac;s similarity conjecture was proposed by P. Erd & odblac;s in 1974. The conjecture remains open for exponentially decaying sequences as well as Cantor sets that have both Newhouse thickness and Hausdorff dimension zero. In this article-written 50 years after the conjecture was first proposed-we review progress on some new variants of the original problem, namely the bi-Lipschitz variant, the topological variant, and a variant "in the large". These problems were recently studied by the authors and their collaborators. Each of them offers new perspectives on the original conjecture.