The board / number theory / #772
Erdős Problem #772
PROVEDdatabase updated 2025-08-31
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
\(H_k(n)\) is the maximal \(r\) such that if \(A\subset\mathbb{N}\) has \(\lvert A\rvert=n\) and \(\| 1_A\ast 1_A\|_\infty \leq k\) then \(A\) contains a Sidon set of size at least \(r\). Here \(1_A\ast 1_A(m)\) counts the ordered pairs \((a,b)\in A^2\) with \(a+b=m\). (The value is truncated at \(n\), which only matters when no such \(A\) exists.)
Let \(k\geq 1\) and \(H_k(n)\) be the maximal \(r\) such that if \(A\subset\mathbb{N}\) has \(\lvert A\rvert=n\) and \(\| 1_A\ast 1_A\|_\infty \leq k\) then \(A\) contains a Sidon set of size at least \(r\). Is it true that \(H_k(n)/n^{1/2}\to \infty\)? The answer is yes, and in fact \(H_k(n) \gg_k n^{2/3}\), proved by Alon and Erdős [AlEr85].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/772.Attempts
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