Source labels openChecked July 26, 2026

Erdős Problem 865

There exists a constant $C>0$ such that, for all large $N$ , if $A\subseteq \{1,\ldots,N\}$ has size at least $\frac{5}{8}N+C$ then there are distinct $a,b,c\in A$ such that $a+b,a+c,b+c\in A$ .

Combinatorics AMS 5 AMS 11

Mathematical statement

There exists a constant $C>0$ such that, for all large $N$ , if $A\subseteq \{1,\ldots,N\}$ has size at least $\frac{5}{8}N+C$ then there are distinct $a,b,c\in A$ such that $a+b,a+c,b+c\in A$ .

A problem of Erdős and Sós (also earlier considered by Choi, Erdős, and Szemerédi [CES75], but Erdős had forgotten this).

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_865

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_865 :    ∃ C > 0, ∀ᶠ (N : ℕ) in atTop,      ∀ A ⊆ Icc 1 N, A.card ≥ (5 / 8 : ℝ) * N + C →      ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, a ≠ b ∧ a ≠ c ∧ b ≠ c ∧      a + b ∈ A ∧ a + c ∈ A ∧ b + c ∈ A := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization erdosproblems.com/865background