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?
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?
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.