Erdős, P. and Freud, R., On sums of a Sidon-sequence. J. Number Theory 38 (1991), no. 2, 196--205, DOI 10.1016/0022-314X(91)90083-N. The site gives this paper and [Er92c] as the problem's sources and calls it a problem of Erdős and Freud; this paper supplies the displayed constant. Its construction on p. 204, a maximally dense Sidon set $B\subset[1,n/3]$ together with $n-B$, has $(2/\sqrt3+o(1))n^{1/2}$ elements and, by the argument printed for its $[1,n/4]$ version on p. 203, all sums distinct except those equal to $n$; it is the set behind the constant $2/\sqrt3$, and the paper does not ask whether it is optimal. Library home: Erdős and Freud 1991; result page Definition (p. 203).

Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. 15 (1992), 34--50, DOI 10.46298/hrj.1992.125. In §2 (pp. 39--40) Erdős reports the Erdős--Freud construction for sets in which only one sum is represented more than once and proposes $\max k=(1+o(1))\frac{2}{\sqrt3}n^{1/2}$ as the probable truth, which is this problem. Library home: Erdős 1992.