Problem #101: Given $n$ points in $\mathbb{R}^2$, no five of which are on...

---

Progress

There are examples of sets of $n$ points with $\sim n^2/6$ many collinear triples and no four points on a line. Such constructions are given by Burr, Grünbaum, and Sloane [BGS74] and Füredi and Palásti [FuPa84].

Grünbaum [Gr76] constructed an example with $\gg n^{3/2}$ such lines. Erdős speculated this may be the correct order of magnitude. This is false: Solymosi and Stojaković [SoSt13] have constructed a set with no five on a line and at least\[n^{2-O(1/\sqrt{\log n})}\]many lines containing exactly four points.

See also [102] and [669]. A generalisation of this problem is asked in [588].

This problem is Problem 71 on Green's open problems list.