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

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

2008-07-30

arxiv.org/abs/0807.4901v2

Mathematics

Combinatorics

18 pages

Scientific paper

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.

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