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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. For $h\ge2$, $g\ge1$, $F_{h,g}(N)$ is the largest size of a subset of $\{1,\ldots,N\}$ in which every integer has at most $g$ representations $a_1+\cdots+a_h$ with $a_1\le\cdots\le a_h$ (p. 1, formula (1)).

Problem 6 (p. 14), called by the paper the main unsolved question: does

$$c_{h,g}=\lim_{N\to+\infty}\frac{F_{h,g}(N)}{N^{1/h}}\qquad(19)$$

exist? The introduction (p. 3) already notes that it is not known whether $F_{h,g}(N)N^{-1/h}$ converges.

Problem 7 (p. 14) asks to compute $c_{h,g}$ explicitly, or, if the limit does not exist, optimal upper and lower bounds. Problem 8 (p. 14) asks to compute $c_{3,1}$ and $c_{2,2}$ explicitly. After Problem 8 the paper recalls that in its reference [15] (Habsieger and Plagne) the authors conjectured $c_{2,2}=2$.

The paper calls the case $g=1$, $h=2$ solved, and the only solved case (p. 9): $c_{2,1}=1$, from formula (2), $F_{2,1}(N)\sim\sqrt N$ (p. 2). The table on p. 13 gives the best known lower and upper bounds for $F_{h,g}(N)N^{-1/h}$ for $2\le h\le4$ and $1\le g\le6$; for $(h,g)=(2,2)$ it gives $1.5118$ and $2.2913$.

Read depth. Claims checked: Problems 6, 7 and 8, the attribution of the conjecture $c_{2,2}=2$ and the table row were read on pp. 3, 9 and 13-14. These are open problems; there is no proof to check.

Proof pointer

None: open problems.

Dependencies

None.

Bears on

• Problem 863: its hypothesis $\lvert A\rvert\sim c_rN^{1/2}$ for maximal $B_2[r]$ sets in $\{1,\ldots,N\}$ is the existence of $c_{2,r}$ in the paper's notation, which Problem 6 asks about. The paper proves nothing on it and does not discuss difference representations.
• Problem 241: see Problem 9, which asks about $c_{3,1}$ directly.