formal-conjectures · erdos-problems · ams-5 · ams-11
Erdős and Sós conjectured that
$f_k(N)\sim \frac{1}{2}\left(1+\sum_{1\leq r\leq k-2}\frac{1}{4^r}\right) N$,
where $f_k(N)$ is the minimal size of a subset of $\{1, \dots, N\}$ guaranteeing $k$ elements
have all pairwise sums in the set.
Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).
Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.
Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://www.erdosproblems.com/865
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/html/2606.29361
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/ErdosProblems/865.lean
Local target: Corpus.Erdos865.erdos_865.variants.sos
Source SHA-256: 817b6472a36402711052e31593fb35d4c2c1575429df9b8c7b60174aadab0578
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.
Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.
Erdős and Sós conjectured that $f_k(N)\sim \frac{1}{2}\left(1+\sum_{1\leq r\leq k-2}\frac{1}{4^r}\right) N$, where $f_k(N)$ is the minimal size of a subset of $\{1, \dots, N\}$ guaranteeing $k$ elements have all pairwise sums in the set.
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