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OP-9339OpenGeometry and topology

Erdős #1086 · geometry / distances

Let g(n)g(n) be minimal such that any set of nn points in R2\mathbb{R}^2 contains the vertices of at most g(n)g(n) many triangles with the same area. Estimate g(n)g(n).

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Problem statement

Let g(n)g(n) be minimal such that any set of nn points in R2\mathbb{R}^2 contains the vertices of at most g(n)g(n) many triangles with the same area. Estimate g(n)g(n).

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Background

Equivalently, how many triangles of area 11 can a set of nn points in R2\mathbb{R}^2 determine? Erdős and Purdy attribute this question to Oppenheim. Erdős and Purdy [ErPu71] proved

n2log⁡log⁡n≪g(n)≪n5/2,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)≪n20/9g(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 gdr(n)g_d^{r}(n) be minimal such that any set of nn points in Rd\mathbb{R}^d contains the vertices of at most gdr(n)g_d^{r}(n) many rr-dimensional simplices with the same volume.

Erdős and Purdy [ErPu71] proved g32(n)≪n8/3g_3^2(n) \ll n^{8/3}, and Dumitrescu, Sharir, and Tóth [DST09] improved this to g32(n)≪n2.4286g_3^2(n) \ll n^{2.4286}. Erdős and Purdy [ErPu71] proved g62(n)≫n3g_6^2(n)\gg n^3. Purdy [Pu74] proved

g42(n)≤g52(n)≪n3−cg_4^2(n)\leq g^2_5(n) \ll n^{3-c}

for some constant c>0c>0. An observation of Oppenheim (using a construction of Lenz) detailed in [ErPu71] shows that

g2k+2k(n)≥(1(k+1)k+1+o(1))nk+1g_{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].

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The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1086 link

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