Problem #195: What is the largest $k$ such that in any permutation of...

What is the largest $k$ such that in any permutation of $\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\cdots

SOLVED

Problem Statement

What is the largest $k$ such that in any permutation of $\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1<\cdots<x_k$?

Categories: Arithmetic Progressions

Geneson [Ge19] proved that $k\leq 5$. Adenwalla [Ad22] proved that $k\leq 4$.

See also [194] and [196].

Source: erdosproblems.com/195 | Last verified: January 14, 2026