Problem 589

Erdős Problem #589: declared status 'open'. Formalized: no. Let $g(n)$ be maximal such that in any set of $n$ points in $\mathbb{R}^2$ with no four points on a line there exists a subset on $g(n)$ points with no three points on a line. Estimate $g(n)$. Current best: A similar question can be asked for a set with no $k$ points on a line, searching for a subset with no $l$ points on a line, for any $3\leq l<k$. F\"{u}eredi [Fu91b] proved$$n^{1/2}\log n\ll g(n)=o(n).$$Balogh and Solymosi [BaSo18] improved the upper bound to$$g(n) \ll n^{5/6+o(1)}.$$ References [BaSo18] Balogh, J\'{o}zsef and Solymosi, J\'{o}zsef, On the number of points in general position in the plane. Prize: no. Tags: geometry.