540.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 540

References:

erdosproblems.com/540
[Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
[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.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[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.

Imports (1)

FormalConjecturesUtil

def Erdos540.HasZeroSubsetSum A finite set has a non-empty subset whose sum is zero line 48 source
Source documentation
A finite set has a non-empty subset whose sum is zero.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: AddCommMonoid, Finset
Proof uses: Exists, And, LE.le, Finset.Nonempty, Eq, Finset.sum, id, OfNat.ofNat
8 instances, coercions and auxiliary constants left out.
```
def HasZeroSubsetSum {G : Type*} [AddCommMonoid G] (A : Finset G) : Prop :=
…
```
theorem Erdos540.erdos_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 $\sum_{n\in S}n\equiv 0\pmod{N}$? sorry line 58 source
Source documentation
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.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, Finset, ZMod, LE.le, HMul.hMul, Real.sqrt, Finset.card, Erdos540.HasZeroSubsetSum
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_540 : answer(True) ↔
    ∃ C : ℝ, 0 < C ∧
      ∀ (N : ℕ), 0 < N → ∀ A : Finset (ZMod N),
        C * Real.sqrt N ≤ A.card →
          HasZeroSubsetSum A := by
…
```
Mentions: Erdos540.HasZeroSubsetSum