Topological Erdős similarity conjecture for uncountable sets in non-meager Baire sets

Prove that for every uncountable subset F of the real line, there exists a non-meager Baire subset A of the real line that contains no affine image of F, thereby showing that uncountable sets are not universal in the class of non-meager Baire sets.

Background

The paper studies point configurations inside sets with sufficient topological structure and proves that bounded countable sets are universal in non-meager Baire sets, both in Euclidean spaces and more general topological vector spaces. This contrasts with the classical measure-theoretic setting and motivates a topological analogue of the Erdős similarity conjecture.

The conjecture concerns whether uncountable sets can always be avoided by some non-meager Baire set under affine transformations, i.e., that uncountable sets are not universal in the class of non-meager Baire sets. The authors note partial results: for countable sets universality holds in co-meager sets and, via their main theorem, in non-meager Baire sets; for uncountable sets, Gallagher–Lai–Weber have shown certain Cantor sets are not universal, but the general statement for all uncountable sets remains unresolved.