PROVED
This has been solved in the affirmative.
Let $k,r\geq 2$. Does there exist a set $A\subseteq \mathbb{N}$ that contains no non-trivial arithmetic progression of length $k+1$, yet in any $r$-colouring of $A$ there must exist a monochromatic non-trivial arithmetic progression of length $k$?
Erdős
[Er75b] reported that 'Spencer has recently shown that such a sequence exists', but gives no reference.
This is an arithmetic analogue of the graph theory version
[924].
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #966, https://www.erdosproblems.com/966, accessed 2025-11-16
