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Statement

For a finite or infinite sequence $X=(x_i)_{i\in I}$ of positive integers, $\Sigma(X)$ is the set of sums $\sum_{i\in F}x_i$ over finite $F\subseteq I$, and $X$ is complete when every sufficiently large positive integer lies in $\Sigma(X)$ (p. 1). Write $\varphi=(1+\sqrt5)/2$.

Theorem 1 (p. 1). There is a strictly increasing sequence $A=(a_n)_{n\ge1}$ of positive integers such that

• (1) $A\setminus B$ is complete for every finite subsequence $B$ of $A$;
• (2) $A\setminus B$ is incomplete for every infinite subsequence $B$ of $A$;
• (3) $a_{n+1}/a_n\ge 6/5$ for every $n$;
• (4) for some increasing sequence $(N_j)$,

$$\frac{a_{N_j}}{a_{N_j-1}}\longrightarrow\varphi, \qquad \frac{a_{N_j+1}}{a_{N_j}}\longrightarrow\varphi+\frac14 .$$

In particular the sequence $(a_{n+1}/a_n)$ does not converge.

The paper's introduction (p. 1) states the question of Erdős and Graham ([1, p. 57] of the paper): whether a sequence with deletion properties (1) and (2) and $a_{n+1}/a_n\ge1+\varepsilon$ for some $\varepsilon>0$ must satisfy $a_{n+1}/a_n\to\varphi$. Its abstract (p. 1) says that the construction gives a negative answer to Erdős Problem 346.

Source. GPT Pro, A counterexample to Erdős Problem 346, preprint (2026), 5 pp.; Theorem 1 on p. 1, its proof in Section 3, pp. 3--5. The edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The proof was read for structure only.

Proof pointer

Section 3, pp. 3--5. Start from $a_1,\dots,a_4=1,2,3,5$, write $S_n=a_1+\dots+a_n$, and for $n\ge4$ choose integers $b_n$ with $0\le b_n\le a_n/4$ and $a_{n+1}=S_{n-1}+b_n$ (the paper's (3.1), p. 3); such sequences are called admissible and are strictly increasing. Lemma 3 (p. 3) shows that deleting any infinite subsequence from an admissible sequence leaves an incomplete one, which is part (2). The unperturbed choice $b_n=(1-(-1)^n)/2$ makes $x_n=a_{n+1}$ satisfy Graham's recurrence, so Lemma 2 applies to it (p. 4). Lemma 4 (p. 4) gives a finite interval certificate that keeps a tail complete under every later admissible choice, and Lemma 5 (p. 4) shows that an unperturbed continuation eventually yields such certificates for any finitely many tails. The sequence takes $b_4=0$, $b_5=1$, runs unperturbed stretches up to indices $N_j$ chosen so that the tails from $a_m$, $m\le j$, are certified and the ratios $a_{N_j}/a_{N_j-1}$ and $S_{N_j-1}/a_{N_j}$ lie within $1/j$ of $\varphi$, and then sets $b_{N_j}=\lfloor a_{N_j}/4\rfloor$ (the paper's (3.7), p. 5). This gives part (1), the bound $6/5$ from the initial quotients and (3.2), and the two limits in part (4) (p. 5).

Dependencies

Lemma 2 (p. 1) and Lemmas 3, 4 and 5 of the same paper (pp. 3--4).

Bears on

• Problem 346: the theorem gives a sequence with both deletion properties and $a_{n+1}/a_n\ge6/5$ for every $n$ whose ratios $a_{n+1}/a_n$ do not converge, so these hypotheses do not force $a_{n+1}/a_n\to\varphi$. It does not address sequences whose ratios are assumed to converge.