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declared disproved (Lean) · formalizederdos:198
∀ (V : Type u_1) [inst : AddCommGroup V] [Module ℚ V], ∃ X, (∀ (Y : Set V), Y.IsAPOfLength ⊤ → (X ∩ Y).Nonempty) ∧ ∀ (Y : Set V), Y.IsAPOfLength 3 → (X ∩ Y).ncard ≤ 2
additive combinatoricssidon setsarithmetic progressionsN/A5 sources
declared disproved (Lean) · formalizederdos:328
False ↔ ∀ (C : ℕ), 0 < C → ∃ t, ∀ (A : Set ℕ), (∀ (n : ℕ), AdditiveCombinatorics.sumRep A n ≤ C) → ∃ P, ⋃ i, P i = A ∧ Set.univ.PairwiseDisjoint P ∧ ∀ (i : Fin t) (n : ℕ), AdditiveCombinatorics.sumRep (P i) n < C
number theoryadditive combinatoricsN/A4 sources
declared disproved (Lean) · formalizederdos:331
False ↔ ∀ (A B : Set ℕ), ((fun n => ↑n ^ (1 / 2)) =O[Filter.atTop] fun n => ↑(Nat.count A n)) → ((fun n => ↑n ^ (1 / 2)) =O[Filter.atTop] fun n => ↑(Nat.count B n)) → {(a₁, a₂, b₁, b₂) | a₁ ∈ A ∧ a₂ ∈ A ∧ b₁ ∈ B ∧ b₂ ∈ B ∧ a₁ ≠ a₂ ∧ a₁ + b₂ = a₂ + b₁}.Infinite
number theoryadditive combinatoricsN/A5 sources
declared disproved (Lean) · formalizederdos:337
False ↔ ∀ (A : Set ℕ), A.IsAsymptoticAddBasis → ((fun N => ↑(A ∩ Set.Icc 1 N).ncard) =o[Filter.atTop] fun N => ↑N) → Filter.Tendsto (fun N => ↑((A + A) ∩ Set.Icc 1 N).ncard / ↑(A ∩ Set.Icc 1 N).ncard) Filter.atTop Filter.atTop
number theoryadditive combinatoricsadditive basisN/A5 sources
declared disproved (Lean) · formalized · $1000erdos:707
Erdős conjectured, in over a dozen papers spanning 1976 to 1997 and with a 1000 dollars prize attached, that every finite Sidon set extends to a perfect difference set modulo p2+p+1p^2+p+1 for some prime pp. Alexeev and Mixon establish that {1,2,4,8}\{1,2,4,8\} is a counterexample - and discovered along the way that Marshall Hall, Jr. had published a different counterexample three decades before Erdős first posed the problem, unnoticed by the community for half a century.
additive combinatoricssidon setsN/A6 sources
declared disproved (Lean) · formalizederdos:867
False ↔ ∃ C, ∀ (N : ℕ), ∀ A ⊆ Finset.Icc 1 N, Erdos867.ConsecutiveSumFree A → ↑A.card ≤ ↑N / 2 + C
additive combinatoricspossible5 sources

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