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Green and Tao determine $f(n)$ exactly for all sufficiently large $n$: $n$ points in the real plane, not all on one line, determine at least $n/2$ ordinary lines, lines through exactly two of the points (Theorem 1.2, the Dirac-Motzkin conjecture), and more precisely at least $f(n)$ of them, where $f(2m)=m$, $f(4m+1)=3m$ and $f(4m-1)=3m-3$, so $f(n)=3\lfloor n/4\rfloor$ for odd $n$ (Theorem 2.2). Every one of these values is attained: $n/2$ equally spaced points on a circle together with the $n/2$ points at infinity in the directions of the lines joining them (the sides and diagonals of the regular polygon) span exactly $n/2$ ordinary lines. When $n/2$ is even, adding the center gives $n+1$ points with $3n/4$ ordinary lines, and removing a point at infinity that lies on the tangents at two opposite circle points gives $n-1$ points with $3n/4-3$; when $n/2$ is odd, removing any one point at infinity gives $n-1$ points with $3(n-2)/4$ (Proposition 2.1, the Böröczky examples); Theorem 2.2 adds that, up to a projective transformation, these are the only sets attaining $f(n)$. Both theorems follow from a structure theorem (Theorems 1.4 and 1.5): a set with at most $Kn$ ordinary lines has all but $O(K)$ of its points on a cubic curve, for $n$ large in terms of $K$. The source card [[../library/discrete_geometry/green_2013_sets_defining_few_ordinary_lines/_index|records Theorems 1.2 and 2.2, Proposition 2.1 and the structure theorems]].

This settles the second question of Problem 210, how fast $f(n)$ grows: linearly, with $f(n)=n/2$ for even $n$ and $f(n)=3\lfloor n/4\rfloor$ for odd $n$ once $n$ is large, after the bounds $3n/7$ of Kelly and Moser and $6n/13$ of Csima and Sawyer, and Motzkin's proof that $f(n)\to\infty$. The threshold $n_0$ is not made explicit, and the exact value of $f(n)$ for small $n$ is not what the question asks.

The result is refereed: Ben Green and Terence Tao, On sets defining few ordinary lines, Discrete Comput. Geom. 50 (2013), no. 2, 409-468; the preprint is arXiv:1208.4714, first posted 2012-08-23. The site's curator, T. F. Bloom, marks the problem proved and credits this paper with the bound $n/2$ for large $n$ and with the odd case.