Originally asked to Erd\H{o}s by Bose. Bose and Chowla \cite{BoCh62} provided a construction proving one half of this, namely\[(1+o(1))N^{1/3}\leq f(N).\]The best upper bound known to date is due to Green \cite{Gr01},\[f(N) \leq ((7/2)^{1/3}+o(1))N^{1/3}\](note that $(7/2)^{1/3}\approx 1.519$).

More generally, Bose and Chowla conjectured that the maximum size of $A\subseteq \{1,\ldots,N\}$ with all $r$-fold sums distinct (aside from the trivial coincidences) then\[\lvert A\rvert \sim N^{1/r}.\]This is known only for $r=2$ (see [30]).

This is discussed in problem C11 of Guy's collection \cite{Gu04}.


References


[BoCh62] Bose, R. C. and Chowla, S., Theorems in the additive theory of numbers. Comment. Math. Helv. (1962/63), 141-147.

[Gr01] Green, Ben, The number of squares and {$B_h[g]$} sets. Acta Arith. (2001), 365-390.

[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.