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Updated Oct 7

Problem 1086

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 $g(n)$ .

Status. Open.

Source. erdosproblems.com/1086, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1086, https://www.erdosproblems.com/1086.

References.

[Ap13] R. Apfelbaum, Geometric Incidences and Repeated Configurations. Ph.D. Dissertation, School of Computer Science, Tel Aviv University (2013).
[ApSh10] Apfelbaum, Roel and Sharir, Micha, An improved bound on the number of unit area triangles. Discrete Comput. Geom. (2010), 753-761.
[DST09] Dumitrescu, Adrian and Sharir, Micha and Tóth, Csaba D., Extremal problems on triangle areas in two and three dimensions. J. Combin. Theory Ser. A (2009), 1177-1198.
[ErPu71] Erdős, Paul and Purdy, George, Some extremal problems in geometry. J. Combinatorial Theory Ser. A (1971), 246-252.
[PaSh92] Pach, János and Sharir, Micha, Repeated angles in the plane and related problems. J. Combin. Theory Ser. A (1992), 12-22.
[Pu74] Purdy, George, Some extremal problems in geometry. Discrete Math. (1974), 305-315.
[RaSh17] Raz, Orit E. and Sharir, Micha, The number of unit-area triangles in the plane: theme and variation. Combinatorica 37 (2017), 1221-1240.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (17)

Distance Problems distance_problems/apfelbaum_2010_improved_bound_number_unit_area_triangles Conjecture (p. 9): the unit-area triangle count is nearly quadratic Theorem 2.1: n planar points span O*(n^{9/4}) unit-area triangles distance_problems/dumitrescu_2009_extremal_problems_triangle_areas_two_three Theorem 1: n planar points span O(n^{44/19}) unit-area triangles Theorem 13: n planar lines determine O(n^{7/3}) unit-area triangles, and Omega(n^2) can occur Theorem 2: n points in convex position can span Omega(n log n) unit-area triangles Theorem 3: n planar points span at most (2/3)(n^2 - n) minimum-area triangles Erdős–Purdy: Some extremal problems in geometry Theorem 1 (p. 248): at most 4n^{3/2} equal-area triangles at a fixed vertex, so at most 4n^{5/2} in the plane Theorem 2 (p. 249): n points in the plane can span cn^2 log log n triangles of the same area Theorem 3 (p. 250): at most cn^{2-1/3} equal-area triangles at a fixed vertex in three-space distance_problems/raz_2017_number_unit_area_triangles_plane_theme Theorem 1: n planar points span O(n^{20/9}) unit-area triangles Theorem 11: a convex grid of n points spans O(n^{31/14}) unit-area triangles Theorem 8: points on any three lines can span Theta(n^2) unit-area triangles

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