erdős #1

If $A \subseteq {1, \dots, N}$ with $∣ A ∣ = n$ is such that the subset sums $\sum_{a \in S} a$ are distinct for all $S \subseteq A$ then $N ≫ 2^{n} .$

Open — best to date is a honest null, not yet sealed.

number theory · open · prize $500 · formalized (Lean) · 1 attempt

machinery: Sidon/B_h,additive-combinatorics,Littlewood-Offord,subset-sum-distinct,second-moment-concentration,dissociated-set

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

attempted via frontier 'sidon/B2' (transfer_strength=partial) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'partial'. Do not re-attack cold; needs a new idea or richer accumulated context.

unverified AI candidates (2)

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

What you wrote is *exactly* the classical **Erdős distinct subset sums conjecture** (Erdős dated it to 1931):

llm-hunter · codex 5.2 extra high, gpt 5.2, gpt pro 5.2 · unverified

4 LLM attack(s) recorded (codex 5.2 extra high, gpt 5.2, gpt pro 5.2); unverified.

AMS 5 11 · open (literature)

theorem erdos_1 : ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℕ) (_ : IsSumDistinctSet A N),
    N ≠ 0 → C * 2 ^ A.card < N

open