Orchard-Planting Problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a finite set $P$ of distinct points in the Euclidean plane, let \[ t_3(P)=\#\{\ell:\ \ell\text{ is a straight line and }|P\cap\ell|=3\}. \] Only lines containing exactly three selected points are counted. No general-position assumption or prohibition on lines containing four or more selected points is imposed.

Statement recorded in the supplied source.

For every integer $n\ge1$, determine \[ T_3(n)=\max\{t_3(P):P\subset\mathbb R^2,\ |P|=n\}. \] [S2]

Short English statement

What is the largest number of straight rows containing exactly three trees obtainable by planting n trees at distinct planar points?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1382 (GEOM-023). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ben Green and Terence Tao, On sets defining few ordinary lines (2013), arXiv:1208.4714v3, Theorem 1.3 and Section 9 (the orchard problem). https://arxiv.org/abs/1208.4714v3.