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Source. Problem 5.2, the definition of a maximal Sidon set and Problem 5.3 of Section 5, p. 141, of A. Sárközy and V. T. Sós, On additive representation functions, in R. L. Graham et al. (eds.), The Mathematics of Paul Erdős I, Springer, 1997, 129--150, doi:10.1007/978-3-642-60408-9_11, as identified on the source card.

Statement

Setting (pp. 130 and 141). For $g\in\mathbb N$, $B_2[g]$ is the class of sets $\mathcal A\subset\mathbb N_0$ in which every $n$ has at most $g$ representations $a+a'=n$ with $a,a'\in\mathcal A$, $a\le a'$; the sets in $B_2[1]$ are the Sidon sets. For a Sidon set $\mathcal A\subset\{1,2,\ldots,N\}$, $H(\mathcal A,N,g)$ is the largest cardinality of a set $\mathcal E\in B_2[g]$ with $\mathcal A\subset\mathcal E\subset\{1,2,\ldots,N\}$ (Problem 5.2).

Definition (p. 141). A Sidon set $\mathcal A\subset\{1,2,\ldots,N\}$ is maximal when no $b\in\{1,2,\ldots,N\}\setminus\mathcal A$ leaves $\mathcal A\cup\{b\}$ a Sidon set. The paper adds, in parentheses, "Note that very little is known on the cardinality of maximal Sidon sets; see Problem 15 in [15]", where [15] is P. Erdős and A. Sárközy, Problems and results on additive properties of general sequences, II, Acta Math. Hung. 48 (1986), 201--211.

Problem 5.3 (p. 141, quoted). "Does there exist a maximal Sidon set such that it can be embedded into a much larger set $\mathcal E\in B_2[g]$?" The paper restates it with $L(N,g)=\max(H(\mathcal A,N,g)-|\mathcal A|)$, the maximum taken over all maximal Sidon sets $\mathcal A\subset\{1,2,\ldots,N\}$, and asks whether $\lim_{N\to+\infty}L(N,2)=+\infty$ and whether $\lim_{N\to+\infty}(L(N,g+1)-L(N,g))=+\infty$ for all $g\in\mathbb N$.

The companion Problem 5.2 (p. 141) asks the same about every Sidon set, through $K(N,g)=\min(H(\mathcal A,N,g)-|\mathcal A|)$ over all Sidon sets $\mathcal A\subset\{1,2,\ldots,N\}$: whether $\lim_{N\to+\infty}K(N,2)=+\infty$, how fast $K(N,g)$ grows in $N$, and whether $\lim_{N\to+\infty}(K(N,g+1)-K(N,g))=+\infty$ for all $g\in\mathbb N$. The paper proves nothing about either problem.

Read depth. Claims checked: the definitions and both problems were read clause by clause on the printed page. There is no proof to check.

Proof pointer

None; these are open questions as the paper poses them.

Dependencies

None.

Bears on

• Problem 156: the paper's definition of a maximal Sidon set in $\{1,2,\ldots,N\}$ is the one the problem uses, and its remark records that, when it was written, very little was known on the cardinality of such sets. The question it poses concerns embedding maximal Sidon sets in larger $B_2[g]$ sets, not their least size, and the paper gives no bound on the size of a maximal Sidon set.