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A proof of Green’s conjecture regarding the removal properties of sets of linear equations
"... A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1,..., n} which contains o(n p−ℓ) 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 homo ..."
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Cited by 8 (1 self)
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A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1,..., n} which contains o(n p−ℓ) 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. In this paper we confirm Green’s conjecture by showing that every set of linear equations (even nonhomogenous) has the removal property. We also discuss some applications of our result in theoretical computer science, and in particular, use it to resolve a conjecture of Bhattacharyya, Chen, Sudan and Xie [7] related to algorithms for testing properties of boolean functions.
Removal lemma for infinitelymany forbidden hypergraphs and property testing
, 2008
"... We prove a removal lemma for infinitelymany forbidden hypergraphs. It affirmatively settles a question on property testing raised by Alon and Shapira (2005) [2, 3]. All monotone hypergraph properties and all hereditary partite hypergraph properties are testable. Our proof constructs a constanttim ..."
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Cited by 5 (4 self)
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We prove a removal lemma for infinitelymany forbidden hypergraphs. It affirmatively settles a question on property testing raised by Alon and Shapira (2005) [2, 3]. All monotone hypergraph properties and all hereditary partite hypergraph properties are testable. Our proof constructs a constanttime probabilistic algorithm to edit a small number of edges. It also gives a quantitative bound in terms of a coloring number of the property. It is based on a new hypergraph regularity lemma [14].
Linear Ramsey Numbers for boundeddegree hypergraphs
, 2007
"... We show that the the Ramsey number of every boundeddegree uniform hypergraph is linear with respect to the number of vertices. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. [8] showed in 1983. Our result may demonstrate the potential of a new hypergr ..."
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We show that the the Ramsey number of every boundeddegree uniform hypergraph is linear with respect to the number of vertices. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. [8] showed in 1983. Our result may demonstrate the potential of a new hypergraph regularity lemma by [18].