Erdős Problem 176

### What is the conjecture

https://www.erdosproblems.com/176

Let $N(k,\ell)$ be the minimal $N$ such that for any $f:\\{1,\ldots,N\\}\to\\{-1,1\\}$ there must exist a $k$-term arithmetic progression $P$ such that

$$\left\lvert \sum_{n\in P}f(n)\right\rvert\geq \ell.$$

Find good upper bounds for $N(k,\ell)$. Is it true that for any $c>0$ there exists some $C>1$ such that
$$N(k,ck)\leq C^k?$$
What about
$$N(k,2)\leq C^k$$
or
$$N(k,\sqrt{k})\leq C^k?$$

Status: open

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else