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Polychromatic blocks and explicit lower bounds for multicolour van der Waerden numbers
Let $W(r,k)$ be the least $N$ such that every $r$-colouring of $\{1,\dots,N\}$ contains a monochromatic $k$-term arithmetic progression. Many of the best lower bounds for $W(r,k)$ with $r\ge3$ come from blow-up recurrences of Blankenship, Cummings and Taranchuk and of Xu, in which a known colouring
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