Israel Journal of Mathematics

, Volume 187, Issue 1, pp 193–207 | Cite as

A removal lemma for systems of linear equations over finite fields



We prove a removal lemma for systems of linear equations over finite fields: let X 1, …, X m be subsets of the finite field F q and let A be a (k × m) matrix with coefficients in F q ; if the linear system Ax = b has o(q m−k ) solutions with x i X i , then we can eliminate all these solutions by deleting o(q) elements from each X i . This extends a result of Green [Geometric and Functional Analysis 15 (2) (2005), 340–376] for a single linear equation in abelian groups to systems of linear equations. In particular, we also obtain an analogous result for systems of equations over integers, a result conjectured by Green. Our proof uses the colored version of the hypergraph Removal Lemma.


Abelian Group Arithmetic Progression Colored Version Combinatorial Proof Regularity Lemma 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    T. Austin and T. Tao, On the testability and repair of hereditary hypergraph properties, Random Structures Algorithms 36 (2010), 373–463.MathSciNetMATHGoogle Scholar
  2. [2]
    P. Candela, Developments at the interface between combinatorics and Fourier analysis, Ph. D. Thesis, University of Cambridge, 2009.Google Scholar
  3. [3]
    P. Erdos, P. Frankl and V. Rödl, The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs abd Combinatorics 2 (1986), 113–121.CrossRefGoogle Scholar
  4. [4]
    P. Frankl, V. Rödl, Extremal problems on set systems, Random Structures & Algorithms 20 (2002), 131–164.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    W. T. Gowers, Hypergraph regularity and the multidimensional Szemerédi theorem, Annals of Mathematics (2) 166 (2007), 897–946.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    B. Green, A Szemerédi-type regularity lemma in abelian groups, with applications, Geometric and Functional Analysis 15 (2005), 340–376.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    J. Komlós and M. Simonovits, Szemerédi’s regularity lemma and its applications in graph theory, in Combinatorics, Paul Erdős is Eighty, Vol. 2 (Keszthely, 1993), Bolyai Soc. Math. Stud., 2, János Bolyai Math Soc., Budapest, 1996, pp. 295–352.Google Scholar
  8. [8]
    J. Komlós, A. Shokoufandeh, M. Simonovits and E. Szemerédi, The regularity lemma and its applications in graph theory, in Theoretical Aspects of Computer Science (Tehran, 2000), Lecture Notes in Comput. Sci., 2292, Springer, Berlin, 2002, pp. 84–112.Google Scholar
  9. [9]
    D. Král’, O. Serra and L. Vena, A removal lemma for linear systems over finite fields, in Proc. VI Jornadas Matemàtica Discreta y Algorítmica, Ediciones y Publicaciones de la UdL, 2008, pp. 417–424.Google Scholar
  10. [10]
    D. Král’, O. Serra and L. Vena, A combinatorial proof of the removal lemma for groups, Journal of Combinatorial Theory. Series A. 116 (2009), 971–978.MathSciNetCrossRefGoogle Scholar
  11. [11]
    B. Nagle, V. Rödl and M. Schacht, The counting lemma for regular k-uniform hypergraphs, Random Structures & Algorithms 28 (2006), 113–179.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    V. Rödl and J. Skokan, Applications of the regularity lemma for uniform hypergraphs, Random Structures & Algorithms 28 (2006), 180–194.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    I. Z. Ruzsa and E. Szemerédi, Triple systems with no six points carrying three triangles, in Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, Colloq. Math. Soc. János Bolyai, 18, North-Holland, Amsterdam-New York, 1978, pp. 939–945.Google Scholar
  14. [14]
    A. Shapira, Green’s conjecture and testing linear-invariant properties, in Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, 31 May–2 June, 2009 (M. Mitzenmacher eds., ACM, New York, 2009).Google Scholar
  15. [15]
    A. Shapira, A proof of Green’s conjecture regarding the removal properties of sets of linear equations, Journal of the London Mathematical Society. Second Series 81 (2010), 355–373.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    B. Szegedy, The symmetry preserving Removal Lemma, Proceedings of the American Mathematical Society 138 (2010), 405–408.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica 27 (1975), 299–345.Google Scholar
  18. [18]
    T. Tao, A variant of the hypergraph removal lemma, Journal of Combinatorial Theory. Series A 113 (2006), 1257–1280.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    P. Varnavides, On certain sets of positive density, Journal of the London Mathematical Society 34 (1959), 358–360.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2012

Authors and Affiliations

  1. 1.Institute for Theoretical Computer Science (ITI), Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  2. 2.Departament de Matemàtica Aplicada IVUniversitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations