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Statement

A Sidon set in

\{1, \ldots, N\}

has all pairwise sums distinct. Its maximum size is

N^{1/2} + E(N)

; determine the true order of the error

E(N)

(conjectured

O(N^{\varepsilon})

for every

\varepsilon > 0

).

Current frontier

erdosproblems.com/30, OPEN. Singer’s construction gives

(1+o(1))N^{1/2}

; the error satisfies

-O(N^{5/16}) \lesssim E(N) \leq N^{1/4} + 1

(upper bound Erdős–Turán 1941 / Lindström 1969; lower bound Chowla–Erdős, from Singer). Whether

E(N) = O(N^{\varepsilon})

is open.

When this counts as solved

QUANTITATIVE. PROOF_COMPLETE for the true order of

E(N)

with matching bounds (notably settling the

O(N^{\varepsilon})

question). BREAKTHROUGH for any rigorous improvement to the

N^{1/4}

upper or

N^{5/16}

lower error bound.

Classification

quantitative