Problem 169

Erdős Problem #169: declared status 'open'. Formalized: no. 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 $f(k)$. Is$$\lim_{k\to \infty}\frac{f(k)}{\log W(k)}=\infty$$where $W(k)$ is the van der Waerden number? Current best: The current record for $f(3)$ is $f(3)\geq 3.00849$, due to Wr\'{o}blewski [Wr84]. Walker [Wa25] has shown that it suffices to consider Kempner sets (that is, sets of integers defined as all those whose base $b$ digits are contained in some $S\subset \{0,\ldots,b-1\}$ for fixed $b$ and $S$), in the sense that for any $k\geq 3$ and $\epsilon>0$ there is a Kempner set $A$ lacking $k$-term arithmetic progressions such that$$\sum_{n\in A}\frac{1}{n}\geq f(k)-\epsilon.$$ References [Be68] Berlekamp, E. Prize: no. OEIS: A005346. Tags: additive combinatorics, arithmetic progressions.