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Erdős problem 43

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If Sidon sets A,B⊆{1,…,N}A, B \subseteq \{1, \dots, N\} satisfy (A−A)∩(B−B)={0}(A-A) \cap (B-B) = \{0\}, must (∣A∣2)+(∣B∣2)≤(f(N)2)+O(1)\binom{|A|}{2} + \binom{|B|}{2} \le \binom{f(N)}{2} + O(1), where f(N)f(N) is the largest Sidon-set size in [N][N] - and can the bound be improved by a fixed proportion when ∣A∣=∣B∣|A| = |B|?

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Retained declaration

FormalConjectures/ErdosProblems/43.lean

Formal Conjectures

FormalConjectures/ErdosProblems/43.leanErdos43.erdos_43.parts.i9 linesExact file
False ↔  ∃ C,    ∀ᶠ (N : ℕ) in Filter.atTop,      ∀ (A B : Finset ℕ),        A ⊆ Finset.Icc 1 N →          B ⊆ Finset.Icc 1 N →            IsSidon ↑A →              IsSidon ↑B →                (A - A) ∩ (B - B) = {0} → ↑(A.card.choose 2 + B.card.choose 2) ≤ ↑((Erdos43.f N).choose 2) + C
SolvedStatement only, no proof

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Reported by these sources. Not reviewed here.

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