Let $c > 0$ and $n$ be some large integer. What is the size of the largest $A \subseteq {1, \dots, ⌊ c n ⌋}$ such that $n$ is not a sum of a subset of $A$ ? Does this depend on $n$ in an irregular way?

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

number theory · open · possible · formalized (Lean) · 1 attempt

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

attempted via frontier 'additive-basis' (transfer_strength=none) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'none'. 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

Let [ m=\lfloor cn\rfloor,\qquad [m]:={1,2,\dots,m}, ] and write (\Sigma(A)={\sum_{a\in B}a: B\subseteq A}) for the set of subset–sums. We want [ F_c(n):=\max{|A|:A\subseteq [m],\ n\notin \Sigma(A)}. ]

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_361.bigO
    (c : ℝ) (hc : 0 < c)
    (A : ℕ → ℕ)
    (hA : ∀ c n, A n = ((Finset.Icc 1 ⌊c * n⌋₊).powerset.filter
      (fun B ↦ n ≠ ∑ a ∈ B, a)).sup Finset.card) :
    (fun n ↦ (A n : ℝ)) =O[atTop] (answer(sorry) : ℕ → ℝ)

formal-conjectures/361.lean ↗

status

open

evidence

formal

status

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