Problem 1086

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 $g(n)$ .

Source: erdosproblems.com/1086

Geometry
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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.

Problem 1086

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