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 $A'\subsetneq A$?
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.
View the LaTeX source
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 2026-01-16
