combinatorics / Discrepancy Theory
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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Append-only history
a polynomial bound for N(k,2), stronger than the exponential bound asked for; the two-parameter problem remains open
Research memory
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 $N(k, 2) \le C^k$?
a polynomial bound for N(k,2), stronger than the exponential bound asked for; the two-parameter problem remains open
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