Israel Journal of Mathematics

, Volume 192, Issue 1, pp 143–156 | Cite as

Partitions of nonzero elements of a finite field into pairs

  • R. N. KarasevEmail author
  • F. V. Petrov


In this paper we prove that the nonzero elements of a finite field with odd characteristic can be partitioned into pairs with prescribed difference (maybe, with some alternatives) in each pair. The algebraic and topological approaches to such problems are considered. We also give some generalizations of these results to packing translates in a finite or infinite field, and give a short proof of a particular case of the Eliahou-Kervaire-Plaigne theorem about sum-sets.


Simplicial Complex Chromatic Number Free Action Basic Research Grant Ulam Theorem 
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]
    N. Alon, Combinatorial Nullstellensatz, Combinatorics, Probability and Computing 8 (1999), 7–29.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    N. Alon, Additive Latin transversals, Israel Journal of Mathematics 117 (2000), 125–130.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    B. Bollobás and I. Leader, Sums in the grid, Discrete Mathematics 162 (1996), 31–48.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    A. L. Cauchy, Recherches sur les nombres, Journal de l’École Polytechnique 9 (1813), 99–116.Google Scholar
  5. [5]
    S. Dasgupta, G. Károlyi, O. Serra and B. Szegedy, Transversals of additive Latin squares, Israel Journal of Mathematics 126 (2001), 17–28.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    H. Davenport, On the addition of residue classes, Journal of the London Mathematical Society 10 (1935), 30–32.CrossRefGoogle Scholar
  7. [7]
    F. J. Dyson, Statistical theory of the energy levels of complex systems. I, II, III, Journal of Mathematical Physics 3 (1962), 140–175.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    S. Eliahou and M. Kervaire, Sumsets in vector spaces over finite fields, Journal of Number Theory 71 (1998), 12–39.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    S. Eliahou, M. Kervaire and A. Plaigne, Some extensions of the Cauchy-Davenport theorem, Journal of Number Theory 101 (2003), 338–348.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357–368.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    J. Gunson, Proof of a conjecture of Dyson in the statistical theory of energy levels, Journal of Mathematical Physics 3 (1962), 752–753.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    S. Hell, On the number of Tverberg partitions in the prime power case, European Journal of Combinatorics 28 (2007), 347–355.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    J. Kahn and G. Kalai, A counterexample to Borsuk’s conjecture, American Mathematical Society. Bulletin 29 (1993), 60–62.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    G. Károlyi, The Erdős-Heilbronn problem in Abelian groups, Israel Journal of Mathematics 139 (2004), 349–359.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    D. Kohen and I. Sadofschi, A new approach on the seating couples problem,, 2010.
  16. [16]
    L. Lovász, Kneser’s conjecture, chromatic numbers, and homotopy, Journal of Combinatorial Theory, Series A 25 (1978), 319–324.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    J. Matoušek, Using the Borsuk-Ulam Theorem, Springer-Verlag, Berlin-Heidelberg, 2003.zbMATHGoogle Scholar
  18. [18]
    E. Preissmann and M. Mischler, Seating couples around the King’s table and a new characterization of prime numbers, American Mathematical Monthly 116 (2009), 268–272.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    A. Yu. Volovikov, On the index of G-spaces (Russian), Rossiĭkaya Akademiya Nauk. Matematicheskiĭ Sbornik 191 (2000), 3–22; translation in Sbornik. Mathematics 191 (2000), 1259–1277.MathSciNetCrossRefGoogle Scholar
  20. [20]
    A. Vučić and R. T. Živaljević, Note on a conjecture of Sierksma, Discrete and Computational Geometry 9 (1993), 339–349.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    K. Wilson, Proof of a conjecture of Dyson, Journal of Mathematical Physics 3 (1962), 1040–1043.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    R. Živaljević, Topological methods, in Handbook of Discrete and Computational Geometry (J. E. Goodman and J. O’Rourke, eds.), CRC, Boca Raton, 2004.Google Scholar

Copyright information

© Hebrew University Magnes Press 2012

Authors and Affiliations

  1. 1.Department of MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia
  2. 2.Discrete and Computational Geometry LaboratoryYaroslavl’ State UniversityYaroslavl’Russia
  3. 3.Saint-Petersburg Department of the Steklov MathematicalSaint-PetersburgRussia

Personalised recommendations