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Claim. $f(N)\ge(1+o(1))N^{1/3}$. For every prime power $q$, Bose and Chowla construct $q$ integers whose sums of three (repetition allowed, order ignored) are distinct modulo $q^3-1$. Taking $q$ prime and close to $N^{1/3}$ gives a set in $\{1,\ldots,N\}$ of size $(1+o(1))N^{1/3}$ whose sums $a+b+c$ are all distinct apart from the trivial coincidences. Green [[../library/additive_bases/green_2001_number_squares_b_h_g_sets/_index|The number of squares and $B_h[g]$ sets]] (Section 3) reports the theorem in this form: Bose and Chowla showed that the largest $B_h$ set in $\{1,\ldots,N\}$ has at least $N^{1/h}(1+o(1))$ elements, the case $h=3$ being the bound above.

Covers. The lower half of Problem 241, $\liminf_{N\to\infty}f(N)/N^{1/3}\ge1$. The upper half is open; the best upper bound the site gives is Green's $f(N)\le((7/2)^{1/3}+o(1))N^{1/3}$ [Gr01], which settles neither half.

Depends on. No page of this wiki.

Acceptance. Refereed: R. C. Bose and S. Chowla, Theorems in the additive theory of numbers, Comment. Math. Helv. 37 (1962), no. 1, 141--147. The Crossref record gives the issue as December 1962 and no day, so the page is dated to the first day of that month. The site's commentary credits Bose and Chowla with one half of the asymptotic, but on a problem the site labels OPEN that commentary is not review.