Problem Statement
Let $g(n)$ be minimal such that any set of $n$ points in $\mathbb{R}^2$ contains the vertices of at most $g(n)$ many triangles with the same area. Estimate $g(n)$.
Categories:
Geometry Distances
Progress
Equivalently, how many triangles of area $1$ can a set of $n$ points in $\mathbb{R}^2$ determine? Erdős and Purdy attribute this question to Oppenheim. Erdős and Purdy [ErPu71] proved\[n^2\log\log n \ll g(n) \ll n^{5/2},\]and believed the lower bound to be closer to the truth. The upper bound has been improved a number of times - by Pach and Sharir [PaSh92], Dumitrescu, Sharir, and Tóth [DST09], Apfelbaum and Sharir [ApSh10], and Apfaulbaum [Ap13]. The best known bound is\[g(n) \ll n^{20/9}\]by Raz and Sharir [RaSh17].Erdős and Purdy also ask a similar question about the higher-dimensional generalisations - more generally, let $g_d^{r}(n)$ be minimal such that any set of $n$ points in $\mathbb{R}^d$ contains the vertices of at most $g_d^{r}(n)$ many $r$-dimensional simplices with the same volume.
Erdős and Purdy [ErPu71] proved $g_3^2(n) \ll n^{8/3}$, and Dumitrescu, Sharir, and Tóth [DST09] improved this to $g_3^2(n) \ll n^{2.4286}$.
Erdős and Purdy [ErPu71] proved $g_6^2(n)\gg n^3$. Purdy [Pu74] proved\[g_4^2(n)\leq g^2_5(n) \ll n^{3-c}\]for some constant $c>0$. An observation of Oppenheim (using a construction of Lenz) detailed in [ErPu71] shows that\[g_{2k+2}^k(n)\geq \left(\frac{1}{(k+1)^{k+1}}+o(1)\right)n^{k+1}\]and Erdős and Purdy conjecture this is the best possible.
See also [90] and [755].
Source: erdosproblems.com/1086 | Last verified: January 19, 2026