On Triple Lines and Cubic Curves --- the Orchard Problem revisited

TL;DR

This paper investigates the structure of large point sets with many triple lines, proposing a conjecture that such configurations are related to cubic curves, and provides partial results supporting this idea.

Contribution

It introduces a conjecture linking near-optimal triple line configurations to cubic curves and proves several special cases, advancing understanding of the Orchard problem.

Findings

Maximal triple line count is approximately n(n-3)/6 + 1.

Partial results support the conjecture relating configurations to cubic curves.

Connections to the four-in-a-line problem of Erdős are established.

Abstract

Planar point sets with many triple lines (which contain at least three distinct points of the set) have been studied for 180 years, started with Jackson and followed by Sylvester. Green and Tao has shown recently that the maximum possible number of triple lines for an n element set is [n(n - 3)/6] + 1. Here we address the related problem of describing the structure of the asymptotically near-optimal configurations, i.e., of those for which the number of straight lines, which go through three or more points, has a quadratic (i.e., best possible) order of magnitude. We pose the problem whether such point sets must always be related to cubic curves. To support this conjecture we settle various special cases; some of them are also related to the four-in-a-line problem of Erd\H{o}s.