When $p$ is prime Berlekamp \cite{Be68} has proved $W(p+1)\geq p2^p$. Gowers \cite{Go01} has proved\[W(k) \leq 2^{2^{2^{2^{2^{k+9}}}}}.\]The best general lower bound is $W(k)\gg 2^k$, due to Kozik and Shabanov \cite{KoSh16}.

In \cite{Er81} Erd\H{o}s further asks whether $W(k+1)/W(k)\to \infty$, or $W(k+1)-W(k)\to \infty$.

In \cite{Er80} Erd\H{o}s asks whether $W(k)/2^k\to \infty$, and offers \$500 for a proof or disproof of $W(k)^{1/k}\to \infty$.


References


[Be68] Berlekamp, E. R., A construction for partitions which avoid long arithmetic progressions. Canad. Math. Bull. (1968), 409-414.

[Er80] Erd\H{o}s, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

[Er81] Erd\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.

[Go01] Gowers, W. T., A new proof of Szemer\'{e}di's theorem. Geom. Funct. Anal. (2001), 465-588.

[KoSh16] Kozik, Jakub and Shabanov, Dmitry, Improved algorithms for colorings of simple hypergraphs and
applications. J. Combin. Theory Ser. B (2016), 312--332.