Erdős conjecture on arithmetic progressions

Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics. It states that if the sum of the reciprocals of the members of a set A of positive integers diverges, then A contains arbitrarily long arithmetic progressions.

Formally, the conjecture states that if A is a large set in the sense that

$${\displaystyle \sum _{n\in A}{\frac {1}{n}}\ =\ \infty ,}$$

then A contains arithmetic progressions of any given length, meaning that for every positive integer k there are an integer a and a non-zero integer c such that ${\displaystyle \{a,a{+}c,a{+}2c,\ldots ,a{+}kc\}\subset A}$.