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import FormalConjecturesUtilErdős Problem 42: Maximal Sidon Sets and Disjoint Difference Sets
Reference: erdosproblems.com/42
This problem asks whether maximal Sidon sets can coexist with other Sidon sets that have disjoint difference sets (apart from 0).
open Function Set Filteropen scoped Pointwisenamespace Erdos42
Erdős Problem 42: Let M ≥ 1 and N be sufficiently large in terms of M. Is it true that for every
maximal Sidon set A ⊆ {1,…,N} there is another Sidon set B ⊆ {1,…,N} of size M such that
(A - A) ∩ (B - B) = {0}?
This was proved for all $M$ by GPT 5.5 Pro (prompted by Sandhu), see discussion thread for more details.
@[category research solved, AMS 5 11, formal_proof using lean4 at "https://github.com/Shashi456/erdos-formalizations/blob/main/Erdos/P42/CompactCayley/Proof.lean"]
theorem erdos_42 : answer(True) ↔
∀ M ≥ 1, ∀ᶠ N in atTop, ∀ (A : Set ℕ) (_ : IsMaximalSidonSetIn A N),
∃ᵉ (B : Set ℕ), B ⊆ Set.Icc 1 N ∧ IsSidon B ∧ B.ncard = M ∧
((A - A) ∩ (B - B)) = {0} := ⊢ True ↔
∀ M ≥ 1,
∀ᶠ (N : ℕ) in atTop,
∀ (A : Set ℕ), A.IsMaximalSidonSetIn N → ∃ B ⊆ Icc 1 N, IsSidon B ∧ B.ncard = M ∧ (A - A) ∩ (B - B) = {0}
All goals completed! 🐙A variant asking for explicit bounds on how large N needs to be in terms of M.
This version provides a constructive function f such that for all M ≥ 1 and N ≥ f(M), every maximal Sidon set A ⊆ {1,…,N} has another Sidon set B ⊆ {1,…,N} of size M with disjoint difference sets (apart from 0).
@[category research solved, AMS 5 11,
formal_proof using formal_conjectures at "https://github.com/KitaKen1/erdos-42-constructive-variant/blob/1f82c76be43cb56f22e2f7f792e392d5fb3ff78c/lean/Erdos42Constructive.lean"]
theorem erdos_42.variants.constructive : answer(True) ↔
∃ (f : ℕ → ℕ), ∀ (M N : ℕ) (_ : 1 ≤ M) (_ : f M ≤ N),
∀ (A : Set ℕ) (_ : IsMaximalSidonSetIn A N), ∃ᵉ (B : Set ℕ),
B ⊆ Set.Icc 1 N ∧ IsSidon B ∧ B.ncard = M ∧
((A - A) ∩ (B - B)) = {0} := ⊢ True ↔
∃ f,
∀ (M N : ℕ),
1 ≤ M →
f M ≤ N →
∀ (A : Set ℕ), A.IsMaximalSidonSetIn N → ∃ B ⊆ Icc 1 N, IsSidon B ∧ B.ncard = M ∧ (A - A) ∩ (B - B) = {0}
All goals completed! 🐙
The set {1, 2, 4} is a maximal Sidon set in {1, ..., 4}.
hx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})this:1 = 2 ∧ 3 = 2 ∨ 1 = 2 ∧ 3 = 2⊢ False
hx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})h1:1 = 2h2:3 = 2⊢ Falsehx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})h1:1 = 2h2:3 = 2⊢ False hx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})h1:1 = 2h2:3 = 2⊢ Falsehx1:1 ≤ 3hx2:3 ≤ 4hne1:¬3 = 1hne2:¬3 = 2hne4:¬3 = 4hbad:IsSidon ({1, 2, 4} ∪ {3})h1:1 = 2h2:3 = 2⊢ False All goals completed! 🐙
The difference set of {1, 2, 4} is {0, 1, 2, 3}.
@[category textbook, AMS 5 11]
theorem example_difference_set : ({1, 2, 4} : Set ℕ) - {1, 2, 4} = {0, 1, 2, 3} := ⊢ {1, 2, 4} - {1, 2, 4} = {0, 1, 2, 3}
x:ℕ⊢ x ∈ {1, 2, 4} - {1, 2, 4} ↔ x ∈ {0, 1, 2, 3}
x:ℕ⊢ (∃ x_1, (x_1 = 1 ∨ x_1 = 2 ∨ x_1 = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x_1 - y = x) ↔ x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3
x:ℕ⊢ (∃ x_1, (x_1 = 1 ∨ x_1 = 2 ∨ x_1 = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x_1 - y = x) → x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3x:ℕ⊢ x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3 → ∃ x_1, (x_1 = 1 ∨ x_1 = 2 ∨ x_1 = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x_1 - y = x
x:ℕ⊢ (∃ x_1, (x_1 = 1 ∨ x_1 = 2 ∨ x_1 = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x_1 - y = x) → x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3 a:ℕha:a = 1 ∨ a = 2 ∨ a = 4b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ a - b = 0 ∨ a - b = 1 ∨ a - b = 2 ∨ a - b = 3
b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ 1 - b = 0 ∨ 1 - b = 1 ∨ 1 - b = 2 ∨ 1 - b = 3b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ 2 - b = 0 ∨ 2 - b = 1 ∨ 2 - b = 2 ∨ 2 - b = 3b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ 4 - b = 0 ∨ 4 - b = 1 ∨ 4 - b = 2 ∨ 4 - b = 3 b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ 1 - b = 0 ∨ 1 - b = 1 ∨ 1 - b = 2 ∨ 1 - b = 3b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ 2 - b = 0 ∨ 2 - b = 1 ∨ 2 - b = 2 ∨ 2 - b = 3b:ℕhb:b = 1 ∨ b = 2 ∨ b = 4⊢ 4 - b = 0 ∨ 4 - b = 1 ∨ 4 - b = 2 ∨ 4 - b = 3
⊢ 4 - 1 = 0 ∨ 4 - 1 = 1 ∨ 4 - 1 = 2 ∨ 4 - 1 = 3⊢ 4 - 2 = 0 ∨ 4 - 2 = 1 ∨ 4 - 2 = 2 ∨ 4 - 2 = 3⊢ 4 - 4 = 0 ∨ 4 - 4 = 1 ∨ 4 - 4 = 2 ∨ 4 - 4 = 3 ⊢ 1 - 1 = 0 ∨ 1 - 1 = 1 ∨ 1 - 1 = 2 ∨ 1 - 1 = 3⊢ 1 - 2 = 0 ∨ 1 - 2 = 1 ∨ 1 - 2 = 2 ∨ 1 - 2 = 3⊢ 1 - 4 = 0 ∨ 1 - 4 = 1 ∨ 1 - 4 = 2 ∨ 1 - 4 = 3⊢ 2 - 1 = 0 ∨ 2 - 1 = 1 ∨ 2 - 1 = 2 ∨ 2 - 1 = 3⊢ 2 - 2 = 0 ∨ 2 - 2 = 1 ∨ 2 - 2 = 2 ∨ 2 - 2 = 3⊢ 2 - 4 = 0 ∨ 2 - 4 = 1 ∨ 2 - 4 = 2 ∨ 2 - 4 = 3⊢ 4 - 1 = 0 ∨ 4 - 1 = 1 ∨ 4 - 1 = 2 ∨ 4 - 1 = 3⊢ 4 - 2 = 0 ∨ 4 - 2 = 1 ∨ 4 - 2 = 2 ∨ 4 - 2 = 3⊢ 4 - 4 = 0 ∨ 4 - 4 = 1 ∨ 4 - 4 = 2 ∨ 4 - 4 = 3
All goals completed! 🐙
x:ℕ⊢ x = 0 ∨ x = 1 ∨ x = 2 ∨ x = 3 → ∃ x_1, (x_1 = 1 ∨ x_1 = 2 ∨ x_1 = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x_1 - y = x ⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 0⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 1⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 2⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 3
⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 0 exact ⟨1, ⊢ 1 = 1 ∨ 1 = 2 ∨ 1 = 4 All goals completed! 🐙, 1, ⊢ 1 = 1 ∨ 1 = 2 ∨ 1 = 4 All goals completed! 🐙, ⊢ 1 - 1 = 0 All goals completed! 🐙⟩
⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 1 exact ⟨2, ⊢ 2 = 1 ∨ 2 = 2 ∨ 2 = 4 All goals completed! 🐙, 1, ⊢ 1 = 1 ∨ 1 = 2 ∨ 1 = 4 All goals completed! 🐙, ⊢ 2 - 1 = 1 All goals completed! 🐙⟩
⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 2 exact ⟨4, ⊢ 4 = 1 ∨ 4 = 2 ∨ 4 = 4 All goals completed! 🐙, 2, ⊢ 2 = 1 ∨ 2 = 2 ∨ 2 = 4 All goals completed! 🐙, ⊢ 4 - 2 = 2 All goals completed! 🐙⟩
⊢ ∃ x, (x = 1 ∨ x = 2 ∨ x = 4) ∧ ∃ y, (y = 1 ∨ y = 2 ∨ y = 4) ∧ x - y = 3 exact ⟨4, ⊢ 4 = 1 ∨ 4 = 2 ∨ 4 = 4 All goals completed! 🐙, 1, ⊢ 1 = 1 ∨ 1 = 2 ∨ 1 = 4 All goals completed! 🐙, ⊢ 4 - 1 = 3 All goals completed! 🐙⟩For any maximal Sidon set, the difference set contains 0.
A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅⊢ IsSidon {1}
exact fun _ hi _ hj _ hk _ hl _ => A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅x✝⁴:ℕhi:x✝⁴ ∈ {1}x✝³:ℕhj:x✝³ ∈ {1}x✝²:ℕhk:x✝² ∈ {1}x✝¹:ℕhl:x✝¹ ∈ {1}x✝:x✝⁴ + x✝² = x✝³ + x✝¹⊢ x✝⁴ = x✝³ ∧ x✝² = x✝¹ ∨ x✝⁴ = x✝¹ ∧ x✝² = x✝³
A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅x✝⁴:ℕx✝³:ℕx✝²:ℕx✝¹:ℕx✝:x✝⁴ + x✝² = x✝³ + x✝¹hi:x✝⁴ = 1hj:x✝³ = 1hk:x✝² = 1hl:x✝¹ = 1⊢ x✝⁴ = x✝³ ∧ x✝² = x✝¹ ∨ x✝⁴ = x✝¹ ∧ x✝² = x✝³
A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅x✝³:ℕx✝²:ℕx✝¹:ℕhj:x✝³ = 1hk:x✝² = 1hl:x✝¹ = 1x✝:1 + x✝² = x✝³ + x✝¹⊢ 1 = x✝³ ∧ x✝² = x✝¹ ∨ 1 = x✝¹ ∧ x✝² = x✝³
A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅x✝²:ℕx✝¹:ℕhk:x✝² = 1hl:x✝¹ = 1x✝:1 + x✝² = 1 + x✝¹⊢ 1 = 1 ∧ x✝² = x✝¹ ∨ 1 = x✝¹ ∧ x✝² = 1
A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅x✝¹:ℕhl:x✝¹ = 1x✝:1 + 1 = 1 + x✝¹⊢ 1 = 1 ∧ 1 = x✝¹ ∨ 1 = x✝¹ ∧ 1 = 1
A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})hemp:A = ∅x✝:1 + 1 = 1 + 1⊢ 1 = 1 ∧ 1 = 1 ∨ 1 = 1 ∧ 1 = 1
All goals completed! 🐙)
A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})a:ℕha:a ∈ A⊢ 0 ∈ A - A
A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})a:ℕha:a ∈ Athis:a - a ∈ A - A⊢ 0 ∈ A - A
rwa [A:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})a:ℕha:a ∈ Athis:0 ∈ A - A⊢ 0 ∈ A - AA:Set ℕN:ℕhN:1 ≤ NhAsub:A ⊆ Icc 1 NhAsidon:IsSidon AhAmax:∀ ⦃x : ℕ⦄, x ∈ Icc 1 N → x ∉ A → ¬IsSidon (A ∪ {x})a:ℕha:a ∈ Athis:0 ∈ A - A⊢ 0 ∈ A - A at thisend Erdos42