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on Erdős #169 · additive combinatorics / arithmetic progressions

The statement

Let $k\geq 3$ and $f(k)$ be the supremum of $\sum_{n\in A}\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$ -term arithmetic progression. Estimate $f(k)$ .

\lim_{k\to \infty}\frac{f(k)}{\log W(k)}=\infty

where $W(k)$ is the van der Waerden number?

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