In combinatorial mathematics, a large set of positive integers is one such that the infinite sum of the reciprocals diverges. A small set is any subset of the positive integers that is not large; that is, one whose sum of reciprocals converges. Large sets appear in the Müntz–Szász theorem and in the Erdős conjecture on arithmetic progressions.

Examples

Properties

Open problems involving large sets

Paul Erdős conjectured that all large sets contain arbitrarily long arithmetic progressions. He offered a prize of $3000 for a proof, more than for any of his other conjectures, and joked that this prize offer violated the minimum wage law. The question is still open. It is not known how to identify whether a given set is large or small in general. As a result, there are many sets which are not known to be either large or small.