Problem Statement
For what values of $0\leq m<n$ is there a complete sequence $A=\{a_1\leq a_2\leq \cdots\}$ of integers such that
- $A$ remains complete after removing any $m$ elements, but
- $A$ is not complete after removing any $n$ elements?
Categories:
Number Theory Complete Sequences
Progress
The Fibonacci sequence $1,1,2,3,5,\ldots$ shows that $m=1$ and $n=2$ is possible. The sequence of powers of $2$ shows that $m=0$ and $n=1$ is possible. The case $m=2$ and $n=3$ is not known.van Doorn has shown that no such sequence exists for $2\leq m<n$ if we interpret complete in the strong sense that\[\left\{ \sum_{n\in B}n : \textrm{ for all finite }B\subset A\right\}=\mathbb{N}.\]Erdős and Graham most likely meant the weaker notion of completeness which allows finitely many exceptions, however.
Source: erdosproblems.com/348 | Last verified: January 14, 2026