A visit to the Erdős problem (Q1382427)

From author's introduction: On several occasions Erdős has asked the following: {Question:} Is it true that for every infinite set \(A\) of real numbers there exists a measurable set of positive measure that does not contain a subset similar to \(A\)? In this note we first give some history of the papers related to the Erdős question and then show how to transform the question into a finite combinatorial problem. Finally, we point out how to use this combinatorial setting to extend some of the known results on the Erdős question.