Erdős Problem #346

The problem

Let $A=\{1\leq a_1< a_2<\cdots\}$ be a set of integers such that $A\backslash B$ is complete for any finite subset $B$ and $A\backslash B$ is not complete for any infinite subset $B$. (Here 'complete' means all sufficiently large integers can be written as a sum of distinct members of the sequence.)

Is it true that if $a_{n+1}/a_n \geq 1+\epsilon$ for some $\epsilon>0$ and all $n$ then

$$\lim_n \frac{a_{n+1}}{a_n}=\frac{1+\sqrt{5}}{2}?$$

Statement quoted from erdosproblems.com/346, maintained by Thomas Bloom. Original source: [ErGr80,p.57]. Current status there: solved.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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