Brown–Graham–Landman Large Sets Conjecture

Let $S\subset\mathbb{N}$. A set $S$ is 2-large if whenever $\mathbb{N}$ is two-colored, there are arbitrarily long arithmetic progressions with common differences in $S$. It is large if whenever $\mathbb{N}$ is finitely colored, there are arbitrarily long arithmetic progressions with common differences in $S$.

Large Sets Conjecture. If $S$ is 2-large, then $S$ is large.

Every large set is 2-large, so the conjecture asserts that restricting van der Waerden's theorem to two colors is sufficient to obtain the corresponding result for every finite coloring. Brown, Graham, and Landman formulated this conjecture in the 1990s; the supplied source identifies it as open.