Problem 1086

Erdős Problem #1086: declared status 'open'. Formalized: no. 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 $g(n)$. Current best: Erd\H{o}s 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 T\'{o}th [DST09], Apfelbaum and Sharir [ApSh10], and Apfaulbaum [Ap13]. The best known bound is$$g(n) \ll n^{20/9}$$by Raz and Sharir [RaSh17]. 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 Erd\H{o}s and Purdy conjecture this is the best possible. Prize: no. Tags: distances, geometry.