The board / additive combinatorics / #840
Erdős Problem #840
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The constant \(c\) is attained by quasi-Sidon sets: there are sets \(A_N \subseteq \{1, \ldots, N\}\) with \(\lvert A_N\rvert = (1+o(1))cN^{1/2}\) and \(\lvert A_N+A_N\rvert=(1+o(1))\binom{\lvert A_N\rvert}{2}\) as \(N \to \infty\).
Let \(f(N)\) be the size of the largest quasi-Sidon subset \(A\subset\{1,\ldots,N\}\), where we say that \(A\) is quasi-Sidon if \[\lvert A+A\rvert=(1+o(1))\binom{\lvert A\rvert}{2}.\] How does \(f(N)\) grow? Following [Er81h, p. 175], we ask for the largest \(c\) such that there are quasi-Sidon sets \(A \subseteq \{1, \ldots, N\}\) with \(\lvert A\rvert = (1+o(1))cN^{1/2}\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/840.Prizenone
Topicsadditive combinatorics, sidon sets
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/840
Attempts
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exact s(k,n) for all n<=37 plus a proof that Pikhurko's 1.863949... bound is optimal within every mode-by-mode Fourier/triangle-inequality certificate; the asymptotic problem remains open.
PARTIAL