B. Alexeev and D. G. Mixon, Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof. arXiv:2510.19804 (2025); published in Proc. Natl. Acad. Sci. USA 123 (2026), no. 21, e2531760123, DOI.

Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.

Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I, Algorithms Combin. 13, Springer (1997), 47--67; p. 54 states the completion conjecture modulo $p^2+p+1$ for a prime power $p=q^\alpha$, "I now feel this conjecture is perhaps too optimistic", and the weaker $(1+\epsilon)n^2$ conjecture; no prize is printed for it. Library home: Erdős 1997; paged at Erdős 1997, Conjecture (p. 54).

Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section C9 "Packing sums of pairs", pp. 176--177: "Erdős also asks if a Sidon sequence $a_1<a_2<\cdots<a_k$ can be prolonged to a perfect difference set (see C10), i.e., $a_1<a_2<\cdots<a_k<a_{k+1}<\cdots<a_{p+1}=p^2+p+1$ with the differences $a_u-a_v$, $1\le u,v\le p+1$, $u\ne v$, representing every nonzero residue mod $p^2+p+1$ exactly once?", followed by the weaker question whether it can be prolonged with $a_n<(1+o(1))n^2$; no prize is printed for it. Section C10 "Modular difference sets and error correcting codes", p. 181, asks "Can a given finite sequence, which contains no repeated differences, always be extended to form a perfect difference set?" after Singer's existence theorem for prime-power $k$ and the conjecture that no perfect difference set exists otherwise. Library home: Guy 2004.

Hall, Jr., Marshall, Cyclic projective planes. Duke Math. J. (1947), 1079-1090.