Problem detail · source-aware
VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Let $k\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\geq c\log n$ for all large $n$ (where $r(n)$ counts representations of $n$ as a sum of at most $k$ elements of $A$) then $A$ must contain a minimal basis of order $k$? The claimed answer is no, for every $k\geq 3$.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.
The proof was developed via an automated multi-turn scaffold that iteratively queried GPT-5.4 Pro and GPT-5.5 Pro over roughly forty turns, with constructions inspired by the Larsen-Larsen order-2 basis; the author later reworked the k=3 case after community concerns and verified the write-up himself and with GPT-5.5 Pro.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
Verification so far is by the author and by GPT-5.5 model runs he links; a Lean formalization attempt is blocked because the underlying Larsen-Larsen probabilistic construction resists autoformalization. No independent human review.
Correctness: unknown · statement fidelity: unaudited · peer review: none
A total refutation is claimed for all k>=3, building on the Larsen-Larsen resolution of problem #868; erdosproblems.com still lists the problem open
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.