Is it true that for every $n \in \mathbb N$ , $\exists N \in \mathbb N$ such that for any subset $A \subseteq \{1,2,...,N\}$ , either $A$ or $\{1,2,..,N\} \setminus A$ contains an arithmetic progression of length $n$ ?
2026-03-27 01:13:57.1774574037
A question on arithmetic progressions
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Color all the points in $A$ red and color all the rest of the points blue. By Van der Waerden's Theorem, if $N = W(2, n)$, then one of these colors must have an arithmetic progression of length $n$.