OPEN
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
\[N(k,\sqrt{k})\leq C^k?\]
Spencer
[Sp73] has proved that if $k=2^tm$ with $m$ odd then
\[N(k,1)=2^t(k-1)+1.\]
Erdős and Graham write that 'no decent bound' is known even for $N(k,2)$. Probabilistic methods imply that, for every fixed constant $c>0$, we have $N(k,ck)>C_c^k$ for some $C_c>1$.
