Optimization Letters

, Volume 12, Issue 5, pp 1011–1017 | Cite as

An exact result for \((0, \pm \, 1)\)-vectors

  • Peter FranklEmail author
Original Paper


After reviewing some memories and results of a dear friend and great collaborator, Michel-Marie Deza, a result (Theorem 8) is proven that could have very well been a joint paper, should not he have departed under tragical circumstances. This new result determines the maximum possible size of a family of \((0, \pm \, 1)\)-vectors without three vectors adding up to the all-zero vector.


Vectors Inequalities Hypergraphs 


  1. 1.
    Borg, P.: Intersecting systems of signed sets. Electron. J. Combin. 14, research paper R41 (2007)Google Scholar
  2. 2.
    Cameron, P.J., Ku, C.Y.: Intersecting families of permutations. Eur. J. Combin. 24, 881–890 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Deza, M.: Solution d’un problème de Erdős–Lovász. J. Comb. Theory Ser. B 16, 166–167 (1974)CrossRefzbMATHGoogle Scholar
  4. 4.
    Deza, M., Frankl, P.: On the maximum number of permutations with given maximal or minimal distance. J. Comb. Theory Ser. A 22, 352–360 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Deza, M., Frankl, P.: Every large set of equidistant \((0, +1, -1)\)-vectors forms a sunflower. Combinatorica 1, 225–231 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ellis, D., Friedgut, E., Pilpel, H.: Intersecting families of permutations. J. Am. Math. Soc. 24, 649–682 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Erdős, P., Ko, C., Rado, R.: Intersection theorems for systems of finite sets. Q. J. Math. Oxf. 2(12), 313–320 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frankl, P.: The Erdős–Ko–Rado theorem is true for \(n=ckt\). Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. I, pp. 365–375, Colloq. Math. Soc. János Bolyai, 18, North-Holland, Amsterdam–New York (1978)Google Scholar
  9. 9.
    Frankl, P., Füredi, Z.: The Erdős–Ko–Rado theorem for integer sequences. SIAM J. Algebraic Discrete Methods 1, 376–381 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kupavskii, A.: Explicit and probabilistic constructions of distance graphs with small clique numbers and large chromatic numbers. Izvest. Math. 78(N1), 59–89 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ponomarenko, E.I., Raigorodskii, A.M.: New upper bounds for the independence numbers with vertices at \({(1, 0, 1)}^n\) and their applications to the problems on the chromatic numbers of distance graphs. Mat. Zametki 96(N1), 138–147 (2014); English transl. Math. Notes 96(N1), 140–148 (2014)Google Scholar
  12. 12.
    Raigorodskii, A.M.: Borsuk’s problem and the chromatic numbers of some metric spaces. Russ. Math. Surv. 56(N1), 103–139 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wilson, R.M.: The exact bound in the Erdős–Ko–Rado theorem. Combinatorica 4, 247–257 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Rényi InstituteBudapestHungary

Personalised recommendations