number-theory / Additive Combinatorics, Ramsey Theory
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 $k$? Answered in the affirmative.
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Append-only history
Erdős reported in 1975 that Spencer had shown existence but gave no reference; no proof was on record before the AI solution
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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 $k$? Answered in the affirmative.
Erdős reported in 1975 that Spencer had shown existence but gave no reference; no proof was on record before the AI solution
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