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For an asymptotic basis $A$ of order $2$, Larsen considers three properties: (P1) $1_A\ast 1_A(n)\to\infty$; (P2) $A$ is the union of two disjoint asymptotic bases of order $2$; (P3) $A$ contains a minimal asymptotic basis of order $2$. Theorem 1 of the note states that all eight combinations of truth values occur. The combination with (P2) true and (P3) false is a union $A=A_1\cup A_2$ of two disjoint bases that contains no minimal basis, which answers the question no.

All eight cases come from one construction (Theorem 2). On the intervals between $N_k=4^{k+1}$ and $N_{k+1}$, random sets $B_k$ and $C_k$ are chosen so that $B$ and $C$ are disjoint, every large integer other than the $N_i$ has many representations in each of $B$ and $C$ (at least $h(n)=\lfloor 10^{-8}n\rfloor$ representations with bounded ratio of summands), and every representation of $N_{k+1}$ as a sum from $B\cup C$ meets a controlled finite set $F_k$ chosen by a selection rule. The integers $N_{k+1}$ take the part that the single integers $N_n$ play in the Erdős–Nathanson constructions: which subsets of $B\cup C$ remain bases is decided by whether they still represent the $N_i$, and the selection rule and two auxiliary functions are tuned to grant or deny each of (P1), (P2) and (P3). Erdős and Nathanson asked whether (P2) implies (P3) as Question 4 of erdos_1988_partitions_bases_into_disjoint_unions_bases; they had shown that $1_A\ast 1_A(n)>C\log n$ for a large constant $C$ gives both (P2) and (P3). The source card is larsen_2026_three_questions_erdos_nathanson_asymptotic_bases; the related claim for Problem 868 is Larsen and Larsen.

Acceptance. The note was posted to the problem's forum on 2026-01-25 and to arXiv on 2026-03-03 (arXiv:2603.03472, 7 pages, not refereed). The site's curator, T. F. Bloom, labels the problem disproved and records the independence of the three properties in the problem's remarks (page last edited 2026-05-02); that is the reviewed evidence. On 2026-04-25 Przemek Chojecki posted to the thread a write-up, https://www.ulam.ai/research/erdos869.pdf, in which GPT-5.5 Pro streamlines Larsen's construction, and reported that they could not formalize it fully; it restates Larsen's result rather than proving it independently, so it is disclosed here and has no page of its own. A Lean 4 formalization of the construction (Lean v4.33.0), produced with Codex and GPT-5.6 Sol and posted in lean-proofs on 2026-08-17, states the negative answer (not_erdos_869, building on the Problem 868 development) with no axiom declarations; this corpus has not audited its statement and the site's label carries no Lean qualifier, so it is a link here and not formalized evidence.

Depends on. Nothing in this wiki; the construction is self-contained apart from standard probabilistic estimates.