TITLE: Has Erdős conjecture on arithmetic progressions been proved?
QUESTION [7 upvotes]: The conjecture states that if $A$ is a set of natural numbers and $$\sum_{n\in A}\frac1n=\infty,$$
then $A$ contains arbitratily long arithmetic progressions.
I wonder has it been proved?

REPLY [4 votes]: This is open even in the simplest case of arithmetic progressions of lengths three. The best result in this direction, for three-terms, is due to T. Bloom building on work of Sanders, and the paper reviews earlier contributions.
You can also see a recent related MO question for further references.