A Proof of Green's Conjecture Regarding the Removal Properties of Sets of Linear Equations

TL;DR

This paper proves Green's conjecture that all sets of linear equations, including non-homogeneous ones, possess the removal property, extending previous results known only for single homogeneous equations.

Contribution

It establishes that every set of linear equations, whether homogeneous or not, has the removal property, confirming Green's long-standing conjecture.

Findings

All sets of linear equations have the removal property.

The removal property applies to both homogeneous and non-homogeneous systems.

This result generalizes previous known cases to all linear systems.

Abstract

A system of \ell linear equations in p unknowns Mx=b is said to have the removal property if every set S \subseteq {1,...,n} which contains o(n^{p-\ell}) solutions of Mx=b can be turned into a set S' containing no solution of Mx=b, by the removal of o(n) elements. Green [GAFA 2005] proved that a single homogenous linear equation always has the removal property and conjectured that every set of homogenous linear equations has the removal property. We confirm Green's conjecture by showing that every set of linear equations (even non-homogenous) has the removal property.