Let $ϵ > 0$ . Is there some set $A \subset N$ of density $> 1 - ϵ$ such that $a_{1} \dots a_{r} = b_{1} \dots b_{s}$ with $a_{i}, b_{j} \in A$ can only hold when $r = s$ ?Similarly, can one always find a set $A \subset {1, \dots, N}$ with this property of size $\geq (1 - o (1)) N$ ?

unreviewedOpen. Worked here; no verified result yet.

number theory · open · formalized (Lean) · 0 attempts

use this data

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

Evidence

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

This is an old question of Erdős (it appears as **Erdős Problem #786**), and **as of Jan 2026 it is still open**. ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

Formal proof

AMS 11 · open (literature)

theorem erdos_786.parts.i : answer(sorry) ↔ ∀ ε > 0, ε ≤ 1 →
    ∃ (A : Set ℕ) (δ : ℝ), 0 ∉ A ∧ 1 - ε < δ ∧ A.HasDensity δ ∧ A.IsMulCardSet

OEIS1

Connections

F_{k} (N)

A \subseteq {1, \dots, N}

k

A

F_{5} (N) = (1 - o (1)) N

F_{2 k + 1} (N) = (1 - o (1)) N

Check it yourself

One command re-derives this record's receipts on your machine.

vela reproduce examples/erdos-problems

Search Vela