Source labels openChecked July 26, 2026

Erdős Problem 817

Let $k \geq 3$ . Define $g_k(n)$ to be the minimal $N$ such that $\{1, ..., N\}$ contains some $A$ of size $|A| = n$ such that $\langle A\rangle = \left\{\sum_{a \in A} \epsilon_a a : \epsilon_a \in\{0, 1\}\right\}$ contains no non-trivial $k$ -term arith...

Combinatorics AMS 5 AMS 11

Mathematical statement

Let $k \geq 3$ . Define $g_k(n)$ to be the minimal $N$ such that $\{1, ..., N\}$ contains some $A$ of size $|A| = n$ such that

\langle A\rangle = \left\{\sum_{a \in A} \epsilon_a a : \epsilon_a \in\{0, 1\}\right\}

contains no non-trivial $k$ -term arithmetic progression. Estimate $g_k(n)$ . In particular, is it true that

g_3(n) \gg 3^n

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_817

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_817 :    answer(sorry) ↔ (fun n => (3 ^ n : ℝ)) =O[atTop] fun n => (g 3 n : ℝ) := 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/817background