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declared solved (Lean) · formalizederdos:42
True ↔ ∀ M ≥ 1, ∀ᶠ (N : ℕ) in Filter.atTop, ∀ (A : Set ℕ), A.IsMaximalSidonSetIn N → ∃ B ⊆ Set.Icc 1 N, IsSidon B ∧ B.ncard = M ∧ (A - A) ∩ (B - B) = {0}
number theorysidon setsadditive combinatoricsN/A6 sources
declared disproved · formalized · $100erdos:43
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|?
number theorysidon setsadditive combinatoricsA143824 · A227590 · A003022 · possible4 sources
declared proved · formalizederdos:152
True ↔ Filter.Tendsto Erdos152.f Filter.atTop Filter.atTop
sidon setsN/A4 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 provederdos:863
Erdős problem 863
number theorysidon setsadditive combinatoricsN/A3 sources