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Claim. S. A. Burr, B. Grünbaum and N. J. A. Sloane, The orchard problem, Geometriae Dedicata 2 (1974), no. 4, 397--424 (source card), prove in Theorem 1 that for every $n\ge3$ there are $n$ points, placed on a cubic curve and chosen through its group law, with at least $1+\lfloor n(n-3)/6\rfloor$ lines through exactly three of them. A projective transformation moves the points into $\mathbb R^2$ without changing which triples are collinear, so $f_3(n)\ge n^2/6-O(n)$ in the notation of Problem 669. A line through at least three of the points contains at least three of the $\binom n2$ pairs, and no pair lies on two lines, so $F_3(n)\le n(n-1)/6$. Since $f_3(n)\le F_3(n)$, both $f_3(n)$ and $F_3(n)$ are $n^2/6-O(n)$, as the site credits to the paper, and

$$\lim_{n\to\infty}\frac{f_3(n)}{n^2}=\lim_{n\to\infty}\frac{F_3(n)}{n^2}=\frac16.$$

Covers. The instance $k=3$: both limits equal $1/6$. It gives nothing for $k\ge4$.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in Geometriae Dedicata. The site labels the problem OPEN, so its remark crediting the paper is not acceptance.