Monotone non-decreasing sequences of the Euler totient function

Abstract

Let $M(x)$ denote the largest cardinality of a subset of $\{n \in \mathbf{N}: n \leq x\}$ on which the Euler totient function $φ(n)$ is non-decreasing. We show that $M(x) = (1+O(\frac{(\log\log x)^5}{\log x})) π(x)$ for all $x \geq 10$, answering questions of Erdős and Pollack--Pomerance--Treviño. A similar result is also obtained for the sum of divisors function $σ(n)$.