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Source. Roel Apfelbaum and Micha Sharir, An improved bound on the number of unit area triangles, Discrete Comput. Geom. 44 (2010), no. 4, 753--761, doi:10.1007/s00454-010-9265-0; read in the arXiv preprint arXiv:1001.4764v1 (26 January 2010), the Discussion paragraph on p. 9 (unnumbered). The edition is identified on the source card.

Read depth. Claims checked: the paragraph was read clause by clause against the preprint. Nothing here is independently reviewed.

Statement

Conjecture (p. 9, unnumbered, in the Discussion after the proof of Theorem 2.1). The authors write that it is "natural to conjecture that our bound is not tight, and that the true bound is nearly quadratic, perhaps coinciding with the lower bound of [4]", where [4] is Erdős and Purdy, Some extremal problems in geometry, J. Combin. Theory 10 (1971), 246--252.

In the corpus's words: the authors expect that the largest number of unit-area triangles spanned by $n$ points in the plane is smaller than the $O^*(n^{9/4})$ of Theorem 2.1, namely nearly quadratic in $n$, and possibly of the order $n^2\log\log n$ of the Erdős–Purdy lattice construction recalled on p. 1. The paper does not make "nearly quadratic" precise.

Motivation

The same paragraph (p. 9) names the slack the authors see in their proof: the count of matching pairs ignores both the requirement that the third vertex of the resulting triangle lies in the point set and the requirement that the top line through that vertex is $k$-rich. The paper proves no part of the conjecture.

Bears on

• Problem 1086: the conjecture is a guess at the order of $g(n)$ for triangles of a fixed positive area, an order that the problem asks to estimate. It is stated as a conjecture and is neither proved nor refuted in the paper.