Problem Statement
Let $f(n)$ be minimal such that, for any $A=\{a_1,\ldots,a_n\}\subseteq [2,\infty)\cap\mathbb{N}$ of size $n$, in any interval $I$ of $f(n)\max(A)$ consecutive integers there exist distinct $x_1,\ldots,x_n\in I$ such that $a_i\mid x_i$.
Obtain good bounds for $f(n)$, or even an asymptotic formula.
Obtain good bounds for $f(n)$, or even an asymptotic formula.
Categories:
Number Theory
Progress
A problem of Erdős and Surányi [ErSu59], who proved\[(\log n)^c \ll f(n) \ll n^{1/2}\]for some constant $c>0$.See also [708].
Source: erdosproblems.com/709 | Last verified: January 16, 2026