OPEN
This is open, and cannot be resolved with a finite computation.
What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there is a function $\delta:A\to \{-1,1\}$ such that\[\sum_{n\in A}\frac{\delta_n}{n}=0\]and\[\sum_{n\in A'}\frac{\delta_n}{n}\neq 0\]for all non-empty $A'\subsetneq A$?
Disclaimer: The open status of this problem reflects the current belief of the owner of this website. There may be literature on this problem that I am unaware of, which may partially or completely solve the stated problem. Please do your own literature search before expending significant effort on solving this problem. If you find any relevant literature not mentioned here, please add this in a comment.
Adenwalla has observed that a lower bound of\[\lvert A\rvert\geq (1-\tfrac{1}{e}+o(1))N\]follows from the main result of Croot
[Cr01], which states that there exists some set of integers $B\subset [(\frac{1}{e}-o(1))N,N]$ such that $\sum_{b\in B}\frac{1}{b}=1$. Since the sum of $\frac{1}{m}$ for $m\in [c_1N,c_2N]$ is asymptotic to $\log(c_2/c_1)$ we must have $\lvert B\rvert \geq (1-\tfrac{1}{e}+o(1))N$.
We may then let $A=B\cup\{1\}$ and choose $\delta(n)=-1$ for all $n\in B$ and $\delta(1)=1$.
This problem has been
formalised in Lean as part of the
Google DeepMind Formal Conjectures project.
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Formalised statement?
Yes
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Additional thanks to: Sarosh Adenwalla and Hayato Egami
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #319, https://www.erdosproblems.com/319, accessed 2026-04-11
