Problem Statement
Must every permutation of $\mathbb{N}$ contain a monotone 4-term arithmetic progression? In other words, given a permutation $x$ of $\mathbb{N}$ must there be indices with either $i<j<k<l$ or $i>j>k>l$ such that $x_i,x_j,x_k,x_l$ are an arithmetic progression?
Categories:
Arithmetic Progressions
Progress
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].
Source: erdosproblems.com/196 | Last verified: January 14, 2026