Topological Erd\H{o}s similarity conjecture and strong measure zero sets

Abstract

We resolve the topological version of the Erd\H{o}s Similarity conjecture introduced previously by Gallagher, Lai and Weber. We show that a set is topologically universal on ${\mathbb R}$ if and only if it is of strong measure zero. As a result of the fact that the Borel conjecture is independent of the \textsf{ZFC} axiomatic set theory, the existence of an uncountable topologically universal set is independent of the \textsf{ZFC}. Moreover, our results can also be generalized to locally compact Polish groups ${\mathbb G}$. Returning to the measure side, we pose Full-Measure universal Erd\H{o}s Similarity Conjecture with strongly meager sets via the duality of measure and category.