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Claim. The question of Problem 880 has the answer yes for $k=2$ and no for every $k\ge3$. Write $h\times A$ for the set of sums of $h$ pairwise distinct elements of $A$ and $\Delta(X)$ for the largest asymptotic gap $\limsup(x_{i+1}-x_i)$ of a set $X=\{x_1<x_2<\cdots\}$. Theorem 1 of N. Hegyvári, F. Hennecart and A. Plagne, Answer to a question by Burr and Erdős on restricted addition, and related results, Combin. Probab. Comput. 16 (2007), no. 5, 747--756, states: (i) if $A\cup 2A$ contains all sufficiently large integers, then $\Delta(A\cup 2\times A)\le2$, and if $2A$ does, then $\Delta(2\times A)\le2$; (ii) for every $h\ge3$ there is a set $A$ such that $h(\{0\}\cup A)$ contains all sufficiently large integers and $\Delta(A\cup 2\times A\cup\cdots\cup h\times A)=\infty$, and a set $A$ with $hA$ containing all sufficiently large integers and $\Delta(h\times A)=\infty$. A set $A$ is a basis of order $k$ in the problem's sense, every large integer a sum of $k$ or fewer elements, exactly when $k(\{0\}\cup A)$ contains all large integers, and the problem's $B$ is $A\cup 2\times A\cup\cdots\cup k\times A$, so part (i) gives $b_{n+1}-b_n\le2$ for all large $n$ when $k=2$, and part (ii) gives a basis of each order $k\ge3$ with $b_{n+1}-b_n$ unbounded. The claim value is disproved: the question asks whether the gaps are bounded for every basis $A$ of every order $k$, and part (ii) refutes that for each $k\ge3$; the case $k=2$, where the assertion holds, is the part of the question that survives. The site labels the problem PROVED, although its remarks report the answer no for every $k\ge3$; the claim value follows the question as stated, which asks about every order $k$.

Argument. The case $k=2$ is the parity observation the site's remarks also make: an odd element of $A+A$ is a sum of two distinct elements, so when every large integer lies in $A\cup(A+A)$, every large odd integer lies in $A\cup 2\times A$, the problem's $B$, whose gaps are therefore at most $2$ (when $A+A$ alone contains every large integer, $2\times A$ itself contains every large odd integer). For $h\ge3$ the paper gives an explicit set: with $x_0=h$ and $x_{n+1}=(3\cdot2^{h-2}-1)x_n^2+hx_n$, let $A_n=[0,x_n^2)\cup\{2^jx_n^2:0\le j\le h-2\}$ and $A=\{0\}\cup\bigcup_{n\ge0}(x_n+A_n)$; sums of $h$ elements of $A_n$ cover a long initial interval, which makes $A$ a basis of order $h$, while sums of distinct elements miss arbitrarily long stretches. The same example gives the paper's Theorem 3: its $k(h)$, the largest over all $A$ with $hA$ cofinite of the least $k$ with $\Delta(k\times A)<\infty$, satisfies $k(h)\ge 2^{h-2}+h-1$, realized by this example, which is why the answer fails for every $h\ge3$. The source card is hegyvari_2007_answer_question_burr_erdos_restricted_addition; the theorem numbering used here is that of the authors' preprint. This outline is a reading aid, not proof coverage.

Acceptance. The refereed evidence is the journal publication cited above (Combinatorics, Probability and Computing; Crossref gives the issue date 2007-09-01, which is the page's date). The reviewed evidence is the documented acceptance by the catalog erdosproblems.com: its curator, Thomas Bloom, credits both answers to this paper in the problem's remarks, and the page carries the label PROVED (accessed 2026-10-07). The proof is not compiled or reviewed here.

Formalization. A third party formalized both answers: not_erdos_880 in src/latest/ErdosProblems/Erdos880.lean of Boris Alexeev's repository https://github.com/plby/lean-proofs, pinned above at the commit of 2026-08-31 that last touched the file (the file was added on 2026-08-17). Its header names Hegyvári, Hennecart and Plagne as informal authors and Codex and GPT-5.6 Sol as formal authors, and its module comment states their theorem, so it is a formalization of this result and a link on this page, not an independent claim. Its theorem is a conjunction: every infinite set whose least order as an asymptotic basis is $2$ has sums of at most two distinct elements with bounded gaps, and for every $h\ge3$ there is an infinite set of least order $h$ whose sums of at most $h$ distinct elements have unbounded gaps; the file imports Mathlib, contains no sorry and prints the theorem's axioms. This corpus has not built the file or audited its definitions against the problem, so the claim carries no formalized evidence.

Depends on. No page of this wiki.