Large sets containing copies of small sets (Q1374563)

One of the typical combinatorial problems is the question if any sufficiently large set contains a subset of a given size having a given property. The famous theorems in this branch are the Ramsey and van der Waerden ones. The author investigates a problem of this type, where the size and property of the included set is ``to be geometrically similar to a fixed set of real numbers'' (\(X\), \(Y\) subsets of real numbers are similar if there exist real numbers \(b\) and \(c\), \(c\neq 0\), s.t. \(Y=\{ cx+b; x\in X\}\)). And ``sufficiently large'' is investigated as to be uncountable, to be open, second Baire property or of positive Lebesgue measure. The author describes both the related results from the literature and his own ones. The matter relates to an Erdös problem. The paper is written in a quite friendly and comprehensible way. It contains sufficient quotation to the matter.