A problem of Erdős and Purdy. It is not even known if $h_c(n)\geq 5$ (see [101]).

It is easy to see that $h_c(n) \ll_c n^{1/2}$, and Erdős at one point [Er95] suggested that perhaps a similar lower bound $h_c(n)\gg_c n^{1/2}$ holds. Zach Hunter has pointed out that this is false, even replacing $>3$ points on each line with $>k$ points: consider the set of points in $\{1,\ldots,m\}^d$ where $n\approx m^d$. These intersect any line in $\ll_d n^{1/d}$ points, and have $\gg_d n^2$ many pairs of points each of which determine a line with at least $k$ points. This is a construction in $\mathbb{R}^d$, but a random projection into $\mathbb{R}^2$ preserves the relevant properties.

This construction shows that $h_c(n) \ll n^{1/\log(1/c)}$.

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