---

Source. Alain Plagne, Recent progress on finite $B_h[g]$ sets, author's manuscript (no venue or year printed), Section 4 (pp. 13-15), the problems on p. 14, as identified on the source card. The file prints no page numbers; pages are counted from its first page.

Statement

Setting. $F_{3,1}(N)$ is the largest size of a subset of $\{1,\ldots,N\}$ in which every integer has at most one representation $a_1+a_2+a_3$ with $a_1\le a_2\le a_3$ (p. 1, formula (1)), and $c_{3,1}$ is the limit of $F_{3,1}(N)/N^{1/3}$ when it exists (Problem 6).

Problem 9 (p. 14, quoted). "Estimate conjecturally $c_{3,1}$ (is $c_{3,1}=1$ reasonable?) or at least find an efficient algorithm to compute the largest $B_3[1]$ set in $\{1,\ldots,N\}$."

The paper introduces it, after Problem 8, as "the easier problem" (p. 14). Its table (p. 13) gives, for $(h,g)=(3,1)$, the lower bound $1$ from Bose and Chowla and the upper bound $1.5183$ from Green, so $1\lesssim F_{3,1}(N)N^{-1/3}\lesssim(7/2)^{1/3}=1.5182\ldots$ (p. 10; the table rounds it to 1.5183).

Read depth. Claims checked: the problem and the bounds were read on pp. 10, 13 and 14. An open problem; there is no proof to check.

Proof pointer

None: an open problem.

Dependencies

Problem 6 for the definition of $c_{3,1}$.

Bears on

• Problem 241: the problem's $f(N)$ is the paper's $F_{3,1}(N)$, sums of three counted with $a\le b\le c$, and it asks whether $f(N)\sim N^{1/3}$, that is, whether $c_{3,1}$ exists and equals $1$. Problem 9 asks whether $c_{3,1}=1$ is reasonable as a conjecture; the paper proves nothing on it.