OPEN
Must every permutation of $\mathbb{N}$ contain a monotone 4-term arithmetic progression $x_1<x_2<x_3<x_4$?
Davis, Entringer, Graham, and Simmons
[DEGS77] have shown that there must exist a monotone 3-term arithmetic progression and need not contain a 5-term arithmetic progression.
See also [194] and [195].
