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Claim. For $k=3$, $d_1=2$ and $d_2=3$ the answer to Problem 1112 is no, in a form stronger than the question asks: for every sequence of positive integers $1\le r_1<r_2<\cdots$ there is a sequence $B=\{b_1<b_2<\cdots\}$ of positive integers with $b_{i+1}\ge r_ib_i$ for all $i$ such that $(A+A+A)\cap B\ne \emptyset$ for every sequence $A=\{a_1<a_2<\cdots\}$ with $2\le a_{i+1}-a_i\le3$ for all $i$. In particular no single ratio $r$ works, so in the notation of the site's commentary (page last edited 28 December 2025) $r_3(2,3)$ does not exist. This is Theorem 3 of B. Bollobás, N. Hegyvári and G. Jin, On a problem of Erdős and Graham, Discrete Math. 175 (1997), no. 1-3, 253--257, cited as [BHJ97] on the problem page. The paper is not held; the statement is taken from its zbMATH review (Zbl 0894.11005, by Erich Härtter), which defines $\mathcal{D}_{(d,d')}$ as the sequences whose consecutive differences lie in $[d,d']$ and $\mathcal{L}_{(r_i,c_i)}$ as the sequences with $b_{i+1}\ge r_ib_i-c_i$, and states Theorem 3 in the form above, and it agrees with the site's commentary, which records the result in the same varying-ratio form. The same paper's Theorem 1 concerns two summands, outside the problem's range $k\ge3$: when $\sup c_i<\infty$, every $B$ with $b_{i+1}\ge2b_i-c_i$ admits an $A$ with gaps in $[2,3]$ and $(A+A)\cap B=\emptyset$, which with its sharpness gives $r_2(2,3)=2$. Johan Land's full claim on its claim page asserts the same nonexistence, in the same varying-ratio form, for every $d_2\le k$.

Covers. The single triple $(k,d_1,d_2)=(3,2,3)$ of the problem's Statement (precise): no ratio exists there, in the varying-ratio form, so no sequence of ratios growing however fast works either. The claim says nothing about other gap bounds or more summands. Under the universal reading of the site's wording, as one assertion over every triple, this theorem would be a full disproof; the problem page does not adopt that reading.

Depends on. No page of this wiki.

Acceptance. Refereed: Discrete Mathematics 175 (1997), no. 1-3, 253--257, doi:10.1016/S0012-365X(96)00122-7, the DOI linked above; the publisher's record gives the issue date as October 1997, filled to its first day for this page's name. The site's curator, Thomas F. Bloom, credits the result to Bollobás, Hegyvári and Jin [BHJ97] in the problem page's commentary (label OPEN (LEAN), page last edited 28 December 2025); the problem is not marked settled, so the credit is not listed as reviewed. The proof is not checked in this corpus.