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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?

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    I've deleted my answer to this post (that the resolution of this conjecture is likely years away), as it is unhelpful.2012-11-09

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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.