Question: determine the exact van der Waerden numbers \(W(r,k)\): the least \(N\) such that every \(r\)-colouring of \(\{1, \dots, N\}\) contains a monochromatic arithmetic progression of length \(k\). Only scattered small values are known (e.g. \(W(2,6) = 1132\)); \(W(2,7)\) and almost all others are open.
Van der Waerden's 1927 theorem guarantees arbitrarily long monochromatic arithmetic progressions in finite colourings of the integers; the numbers quantify it. Only a handful of exact values are known — the landmark computation is \(W(2,6) = 1132\) (Kouril–Paul) — and upper bounds (Gowers) tower above the lower ones. The gap between the best lower-bound constructions and the true values is among the widest in Ramsey theory.
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