Problem 176

Erdős Problem #176: declared status 'open'. Formalized: no. 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$$N(k,\sqrt{k})\leq C^k?$$ Prize: no. Tags: additive combinatorics, arithmetic progressions, discrepancy.