Problem 840

Let $f(N)$ be the size of the largest quasi-Sidon subset $A\subset\{1,\ldots,N\}$ , where we say that $A$ is quasi-Sidon if

\lvert A+A\rvert=(1+o(1))\binom{\lvert A\rvert}{2}.

How does $f(N)$ grow?

Status

Open

No claim settles this problem.

Er81h: Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.
ErFr91: 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 Definition of a quasi-Sidon sequence, p. 203; the construction of $(2/\sqrt3+o(1))n^{1/2}$ elements, the trivial bound (37), $k\le(2+o(1))n^{1/2}$ , and the unproved $1.98$ in its place, p. 204. Library home: Erdős and Freud 1991; result page Definition (p. 203).
Pi06: Pikhurko, Oleg, Dense edge-magic graphs and thin additive bases. Discrete Math. (2006), 2097-2107.