Polychromatic blocks and explicit lower bounds for multicolour van der Waerden numbers

Source abstract

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 is lengthened by a periodic block. Call an $h$-colouring of $Z_q$ $k$-polychromatic if every $k$-term progression with nonzero common difference meets each of the $h$ colours. The argument behind these recurrences shows that such a colouring gives $W(r,k)>q(W(r-\lceil r/h\rceil,k)-1)$ for all $r\ge h$. We show that cubic, quartic, quintic and sextic residue colourings give $k$-polychromatic blocks with $q$ much larger than $k$ (e.g. $q=1,376,317$ for $h=3$ and $k=28$), and use them to get explicit lower bounds. For three colours and $9\le k\le28$ we improve on every published bound we know of, among them $W(3,9)>4,002,705$ and $W(3,28)>159,594,201,561,100,726$. For $3\le r\le6$ and $9\le k\le28$, $77$ of the $80$ cases improve on what the published bounds give under these recurrences and the other three tie. For $r=5$ and $12\le k\le15$ the best values come from palettes that are cyclic intervals of colours indexed by quintic residues. We checked every block by exhaustive enumeration and every base colouring by computer. We also prove that the block modulus is at most $(W(2,k)-1)/(k-1)$, so the method does not improve the asymptotics. For large $k$ the largest cubic block modulus lies between $k^{4-\varepsilon}$ and $k^2 4^k$.