Erdős problem 176

Let $N(k, \ell)$ be the least $N$ such that every $f : [N] \to \{-1, 1\}$ has a $k$ -term arithmetic progression $P$ with $|\sum_{n \in P} f(n)| \ge \ell$ . In particular, is $N(k, 2) \le C^k$ ?

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Let $N(k, \ell)$ be the least $N$ such that every $f : [N] \to \{-1, 1\}$ has a $k$-term arithmetic progression $P$ with $|\sum_{n \in P} f(n)| \ge \ell$. In particular, is $N(k, 2) \le C^k$?

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