Problem 102

Erdős Problem #102: declared status 'open'. Formalized: no. Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\mathbb{R}^2$ such that there are $\geq cn^2$ lines each containing more than three points, there must be some line containing $h_c(n)$ many points. Estimate $h_c(n)$. Is it true that, for fixed $c>0$, we have $h_c(n)\to \infty$? Current best: It is easy to see that $h_c(n) \ll_c n^{1/2}$, and Erd\H{o}s at one point [Er95] suggested that perhaps a similar lower bound $h_c(n)\gg_c n^{1/2}$ holds. Prize: no. Tags: geometry.