Erdős Problem #160 — Let h(N) be the smallest k such that 1,…,N can be coloured with k colours so that every four-term arithmetic progression must contain at least three distinct colours.

Question

Let h(N) be the smallest k such that 1,…,N can be coloured with k colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate h(N).

Erdős Problem #160 — Let $h(N)$ be the smallest $k$ such that $\{1,\ldots,N\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours.

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