SOLVED

Is it true that if $A\subseteq \mathbb{Z}/N\mathbb{Z}$ has size $\gg N^{1/2}$ then there exists some non-empty $S\subseteq A$ such that $\sum_{n\in S}n\equiv 0\pmod{N}$?

A conjecture of Erdős and Heilbronn. The answer is yes, proved by Szemerédi

[Sz70] (in fact for arbitrary finite abelian groups).

Erdős speculated that perhaps the correct threshold is $(2N)^{1/2}$; this is also a conjecture of Selfridge, and has been proved when $N$ is prime by Balandraud [Ba12].