We introduce a connection between Newhouse thickness and patterns through variant of Schmidt’s game introduced by Broderick, Fishman Simmons. This yields an explicit, robust checkable condition that ensures Cantor set in the real line contains long arithmetic progressions and, more generally, homothetic copies large finite sets.

In this note, we provide a new proof of Magidor's theorem that the least strongly compact cardinal can be the least supercompact cardinal. A classical theorem of Magidor [5] states that it is consistent, relative to the existence of a supercompact cardinal, for the least supercompact cardinal to be the least strongly compact cardinal. There are currently two different proofs of this fact—the or...

In this paper we study spaces in which each compact subset is a Gδ-set and compare them to H. W. Martin’s c-semi-stratifiable (CSS) spaces, i.e. spaces in which compact sets are Gδ-sets in a uniform way. We prove that a (countably) compact subset of a Hausdorff space X is metrizable and a Gδ-subset of X provided X has a δθ-base, or a point-countable, T1-point-separating open cover, or a quasi-G...

Reconstructing a 3D shape from sample points is a central problem faced in medical applications, reverse engineering, natural sciences, cultural heritage projects, etc. While these applications motivated intense research on 3D surface reconstruction, the problem of reconstructing more general shapes hardly received any attention. This paper develops a reconstruction algorithm changing the 3D re...