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Statement

Setting (p. 126). $W(n)$ is the least $W$ such that in any partition of $\{1,2,\ldots,W\}$ into two classes some class contains an $n$-term arithmetic progression.

Conjecture (p. 127, unnumbered, with a prize). For all $n$, $W(n)\le 2^{2^{\cdot^{\cdot^{\cdot^{2}}}}}$, a tower of $n$ twos.

The paper places it after the bounds it reports (p. 126): Shelah's theorem (J. Amer. Math. Soc. 1 (1988)) that $W(n)$ is bounded by a primitive recursive function, displayed as a tower of towers of twos with $n$ layers, and a best known lower bound growing "roughly like $n\cdot 2^n$".

Read depth

Claims checked: the definition, the conjecture and the reported bounds were read on the print. The tower in the print is drawn with $n$ marked along its height.

Dependencies

None in the corpus.

Source. R. L. Graham, Recent trends in Euclidean Ramsey theory, Discrete Math. 136 (1994), 119--127, doi:10.1016/0012-365X(94)00110-5; the edition read is named on the source card.

Bears on

• Problem 138: the problem asks for improved bounds on the two-color van der Waerden number. The paper conjectures a tower-type upper bound and reports the bounds known in 1993; it proves no bound.