AbstractLet M(x) denote the largest cardinality of a subset of $$\{n \in \mathbb {N}: n \le x\}$$ { n ∈ N : n ≤ x } on which the Euler totient function$$\varphi (n)$$ φ ( n ) is nondecreasing. We show that $$M(x) = \left( 1+O\left( \frac{(\log \log x)^5}{\log x}\right) \right) \pi (x)$$ M ( x ) = 1 + O ( log log x ) 5 log x π ( x ) for all $$x \ge 10$$ x ≥ 10 , answering questions of Erdős and Pollack–Pomerance–Treviño. A similar result is also obtained for the sum of divisors function $$\sigma (n)$$ σ ( n ) .