The extensible No-Three-In-Line problem

TL;DR

This paper investigates the infinite grid No-Three-In-Line problem, constructing large dense point sets avoiding collinear triples with near-linear density, and provides computational evidence for even denser configurations.

Contribution

It establishes the existence of large dense collinear-triple-free point sets in the infinite grid, extending classical finite grid results and addressing a question of Erde.

Findings

Existence of point sets with (n/log^{1+ps}n) points avoiding collinear triples.

Computational evidence for sets with at least n/2 points on large grids.

Extension of finite grid results to the infinite grid setting.

Abstract

The classical No-Three-In-Line problem seeks the maximum number of points that may be selected from an $n\times n$ grid while avoiding a collinear triple. The maximum is well known to be linear in $n$. Following a question of Erde, we seek to select sets of large density from the infinite grid $Z^{2}$ while avoiding a collinear triple. We show the existence of such a set which contains $\Theta(n/\log^{1+\varepsilon}n)$ points in $[1,n]^{2}$ for all $n$, where $\varepsilon>0$ is an arbitrarily small real number. We also give computational evidence suggesting that a set of lattice points may exist that has at least $n/2$ points on every large enough $n\times n$ grid.