Corner-free sets via the torus

TL;DR

This paper extends Green-Wolf's ideas to construct larger corner-free sets in the integer grid, improving previous bounds by adapting techniques from 3-term arithmetic progression problems.

Contribution

It introduces a novel approach to constructing corner-free sets in ^2, achieving better bounds by adapting methods from 3-term arithmetic progressions.

Findings

Constructed larger corner-free sets in ^2

Extended Green-Wolf techniques to 2D setting

Achieved improved bounds over previous constructions

Abstract

A corner is a triple of points in $\Bbb{Z}^2$ of the form $(x,y),(x+d,y),(x,y+d)$ where $d\neq 0$. One can think of them as being 2D-analogues to 3-term arithmetic progressions. In this short note, we extend ideas of Green-Wolf from this latter setting to the former, achieving slightly better constructions of corner-free sets.