Let $k\ge 3$ 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$$\underset{k\to \infty }{\lim }\frac{f(k)}{\log W(k)}=\infty$$where $W(k)$ is the van der Waerden number?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

[ f(k)\ :=\ \sup\\{\sum_{n\in A}\frac1n:\ A\subset\mathbb N,\ A\text{ contains no }k\text{-term arithmetic progression}\\}. ]

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.