Problem 348

Erdős Problem #348: declared status 'open'. Formalized: yes. 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 {UL} {LI} $A$ remains complete after removing any $m$ elements, but {/LI} {LI} $A$ is not complete after removing any $n$ elements? {/LI} {/UL} Current best: van Doorn has shown that no such sequence exists for $2\leq m<n$ if we interpret complete in the strong sense that$$\left\{ \sum_{n\in B}n : \textrm{ for all finite }B\subset A\right\}=\mathbb{N}.$$Erd\H{o}s and Graham most likely meant the weaker notion of completeness which allows finitely many exceptions, however. Prize: no. Tags: complete sequences, number theory.