A system of ` linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1, . . . , n} which contains o(np−`) 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 homog...