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Source. Proposition 7, p. 4, of Infinite Deletions from Strongly Minimal Additive Bases, manuscript (2026), no author printed, posted by Svyable in the thread of Erdős Problem 881 on 2026-05-03, https://www.overleaf.com/read/dckvqtggbjzn; the edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the print, and the proof on p. 4 was followed.

Statement

Setting (p. 2). $\mathbb N=\{0,1,2,\ldots\}$; $hA$ is the set of sums of exactly $h$ elements of $A$, repetitions allowed, and $A$ is an asymptotic basis of order $h$ when $hA$ contains every large integer. A basis of order $1$ is thus a set containing $[N_0,\infty)$ for some $N_0$.

Proposition 7 (p. 4). If $A\subset\mathbb N$ is an asymptotic basis of order $1$, then some infinite $B\subset A$ has $A\setminus B$ an asymptotic basis of order $2$.

Proof pointer

P. 4. Take $B$ infinite with $|B\cap[0,x]|=o(x)$, for example elements $b_j\ge2^j$. For large $n$ the choices of $x\in[N_0,n-N_0]$ with $x\in B$ or $n-x\in B$ number $o(n)$, so some $x$ has both $x$ and $n-x$ in $A\setminus B$.

Bears on

• Problem 881: for $k=1$ the paper's hypothesis of strong minimality is not needed; Proposition 7 gives the answer yes for every asymptotic basis of order $1$, read as a set containing all large integers.