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Daniel Larsen and Michael Larsen prove (Theorem 1) that there exist $\varepsilon>0$ and a set $A$ of positive integers such that $r_A(n)>\varepsilon\log n$ for all sufficiently large $n$, where $r_A(n)$ counts the pairs $a\le b$ in $A$ with $a+b=n$ (so also $1_A\ast 1_A(n)>\varepsilon\log n$), and $A$ contains no minimal additive basis of order $2$. Since the representation counts tend to infinity, the first question is answered no; and since the second question asks about an arbitrary fixed $\varepsilon>0$, it is answered no as well, for every $\varepsilon$ below the construction's constant. Erdős and Nathanson had proved the opposite when every large $n$ has more than $c\log n$ representations $n=a+a'$ with $a\le a'$ in $A$, for some $c>1/\log(4/3)$, which the condition $1_A\ast 1_A(n)>\varepsilon\log n$ guarantees only when $\varepsilon>2/\log(4/3)$ (Theorem 2 of erdos_1979_systems_distinct_representatives_minimal_bases_additive), and suggested (pp. 89–90), without conjecturing it formally, that the threshold may not be lowered to every positive constant; the theorem confirms that suggestion. Both questions are answered no, so the claim is a disproof; the exact threshold between the construction's $\varepsilon$ and $1/\log(4/3)$, both measured in pairs $a\le b$, is not determined, and the problem does not ask for it.

The construction is random and proceeds by generations on the intervals $I_n=[2^{2^n},2^{2^{n+1}})$. A random set $A_n\subseteq I_n$ with inclusion probability $\min(1,40\sqrt{\log m/m})$ has representation counts of order $\log m$ (Lemma 2, by Chernoff bounds and Borel–Cantelli). A small set $B_n\subseteq I_n$ of fragile elements is then chosen; every summand of an element of $B_n$ is deleted and replacement elements are added so that each $b\in B_n$ keeps at least $\varepsilon\log b$ representations while every subbasis that still represents $B_n$ is forced to represent the other large integers twice over, with the smallest summand of $B_n$ tending to infinity. A subset $D$ of $A$ that is a basis therefore always has an element whose removal leaves a basis, so no subbasis is minimal. The authors describe the sets $B_n$ as spreading over many elements the role of the single integers $N_n$ of the earlier Erdős–Nathanson construction (erdos_1989_additive_bases_many_representations), which in their words serve as a "canary in the coal mine" (Section 1 of the note); spreading the load is what lets the representation counts grow logarithmically. The source card is larsen_2026_robust_additive_bases_without_minimal_subbases.

Acceptance. The note was posted to the problem's forum on 2026-01-13 (the linked GitHub upload; a revised upload followed on 2026-01-22) and to arXiv on 2026-01-26 (arXiv:2601.18507, 9 pages, not refereed). The site's curator, T. F. Bloom, records the answer in the problem's remarks and labels the problem solved (page last edited 2026-04-03); that is the reviewed evidence. The community database lists the problem as solved, provisionally marked after the forum post, as of its last update, dated 2026-01-13, and records a Lean proof. The site's Lean qualifier refers to a Lean 4 formalization of the note (Lean v4.33.0), produced with Codex and GPT-5.6 Sol and posted in lean-proofs on 2026-08-16, which states the negations of both questions (not_erdos_868 and not_erdos_868_part_ii) with no axiom declarations; this corpus has not audited its statement, 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.