Erdős Problem 540

References:

• erdosproblems.com/540
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• [Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.
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• [Ol68] Olson, John E., An addition theorem modulo {$p$}. J. Combinatorial Theory (1968), 45--52.
• [Ba12] Balandraud, Éric, An addition theorem and maximal zero-sum free sets in $\mathbb{Z}/p\mathbb{Z}$. Israel J. Math. (2012), 405-429.
• [HaZe96] Hamidoune, Yahya Ould and Zémor, Gilles, On zero-free subset sums. Acta Arith. (1996), 143--152.
• [Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
• [ErHe64] Erdős, P. and Heilbronn, H., On the addition of residue classes mod $p$. Acta Arith. (1964), 149--159.
• [Sz70] Szemerédi, E., On a conjecture of Erdős and Heilbronn. Acta Arith. (1970), 227-229.

def Erdos540.HasZeroSubsetSum {G : Type u_1} [AddCommMonoid G] (A : Finset G) : Prop

A finite set has a non-empty subset whose sum is zero.

Equations

• Erdos540.HasZeroSubsetSum A = ∃ S ⊆ A, S.Nonempty ∧ S.sum id = 0

theorem Erdos540.erdos_540 : True ↔ ∃ (C : ℝ), 0 < C ∧ ∀ (N : ℕ), 0 < N → ∀ (A : Finset (ZMod N)), C * √↑N ≤ ↑A.card → HasZeroSubsetSum A

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.