Erdős problem 540

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 $\sum_{n\in S}n\equiv 0\pmod{N}$ ?

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FormalConjectures/ErdosProblems/

540.lean

Retained formal statement1 of 1

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 $\sum_{n\in S}n\equiv 0\pmod{N}$ ?

Szemerédi proved the answer is yes, in fact for arbitrary finite abelian groups.

FormalConjectures/ErdosProblems/540.leanErdos540.erdos_5401 lineExact file

1True ↔ ∃ C, 0 < C ∧ ∀ (N : ℕ), 0 < N → ∀ (A : Finset (ZMod N)), C * √↑N ≤ ↑A.card → Erdos540.HasZeroSubsetSum A

SolvedProof has a holelean4external proof

The proof uses `sorry`: part of the argument is written but not proved. Lean accepts the file; it does not accept the theorem.