The board / additive combinatorics / #877
Erdős Problem #877
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).
`A` is a maximal sum-free subset of `{1, …, n}`.
`fm n` counts the number of maximal sum-free subsets \(A\subseteq\{1,\ldots,n\}\).
Let \(f_m(n)\) count the number of maximal sum-free subsets \(A\subseteq\{1,\ldots,n\}\) - that is, there are no solutions to \(a=b+c\) in \(A\) and \(A\) is maximal with this property. Estimate \(f(n)\) - is it true that \(f_m(n)=o(2^{n/2})\)? A problem of Cameron and Erdős. Łuczak and Schoen [LuSc01] proved that there exists a constant \(c<1/2\) such that \(f_m(n)<2^{cn}\), resolving this question. Balogh, Liu, Sharifzadeh, and Treglown [BLST15] proved that \(f_m(n)=2^{(\frac{1}{4}+o(1))n}\). See [748](https://www.erdosproblems.com/748) for the non-maximal case.
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/877.Prizenone
Topicsadditive combinatorics
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA121269
Original pageerdosproblems.com/877
Attempts
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