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Claim. B. Green and T. Tao, On sets defining few ordinary lines, Discrete & Computational Geometry 50 (2013), no. 2, 409--468 (source card), prove in Theorem 1.3 that there is an $n_0$ such that every set of $n\ge n_0$ points in the plane has at most $\lfloor n(n-3)/6\rfloor+1$ lines containing exactly three of its points. The cubic-curve construction recorded on the Burr--Grünbaum--Sloane claim page attains this number, so in the notation of Problem 669 $f_3(n)=\lfloor n(n-3)/6\rfloor+1$ for all large $n$. The proof rests on a structure theorem: a set with few ordinary lines lies, up to a bounded number of points, on a cubic curve.

Covers. The exact value of $f_3(n)$ for all large $n$. It gives no exact value of $F_3(n)$ and nothing for $k\ge4$.

Depends on. Burr, Grünbaum and Sloane's claim, for the matching construction.

Acceptance. Refereed: the paper appeared in Discrete & Computational Geometry. The site's page does not cite it, and the site labels the problem OPEN, so no curator review is listed.