Skip to main content
SpringerLink
Log in

On a sequence of random distance graphs subject to the zero-one law

  • Large Systems
  • Published:
Problems of Information Transmission Aims and scope Submit manuscript

Abstract

It is known that an Erdös-Rényi random graph obeys a zero-one law for first-order properties. The study of these laws started in 1969 with the work of Yu.V. Glebskii, D.I. Kogan, M.I. Liogon’kii, and V.A. Talanov. We proved in our previous works that a random distance graph does not obey the zero-one law. In this paper a sequence of random distance graphs obeying the zero-one law is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Glebskii, Yu.V., Kogan, D.I., Liogon’kii, M.I., and Talanov, V.A., Volume and Fraction of Satisfiability of Formulas of the Lower Predicate Calculus, Kibernetika (Kiev), 1969, no. 2, pp. 17–26.

  2. Fagin, R., Probabilities on Finite Models, J. Symbolic Logic, 1976, vol. 41, no. 1, pp. 50–58.

    Article  MathSciNet  MATH  Google Scholar 

  3. Zhukovskii, M.E., Zero-One Laws for First-Order Formulas with a Bounded Quantifier Depth, Dokl. Ross. Akad. Nauk, 2011, vol. 436, no. 1, pp. 14–18 [Dokl. Math. (Engl. Transl.), 2011, vol. 83, no. 1, pp. 8–11].

    MathSciNet  Google Scholar 

  4. Kolchin, V.F., Sluchainye grafy (Random Graphs), Moscow: Fizmatlit, 2004, 2nd ed.

    Google Scholar 

  5. Bollobás, B., Random Graphs, Cambridge: Cambridge Univ. Press, 2001, 2nd ed.

    Book  MATH  Google Scholar 

  6. Janson, S., _Luczak, T., and Ruciński, A., Random Graphs, New York: Wiley, 2000.

    Book  MATH  Google Scholar 

  7. Alon, N. and Spencer, J.H., The Probabilistic Method, New York: Wiley, 2000, 2nd ed. Translated under the title Veroyatnostnyi metod, Moscow: BINOM, 2007.

    Book  MATH  Google Scholar 

  8. Uspensky, V.A., Vereshchagin, N.K., and Plisko, V.E., Vvodnyi kurs matematicheskoi logiki (Introductory Course on Mathematical Logic), Moscow: Moscow State Univ., 1997.

    Google Scholar 

  9. Vereshchagin, N.K. and Shen, A., Yazyki i ischisleniya (Languages and Calculi), Moscow: MCCME, 2000.

    Google Scholar 

  10. Shelah, S. and Spencer, J.H., Zero-One Laws for Sparse Random Graphs, J. Amer. Math. Soc., 1988, vol. 1, no. 1, pp. 97–115.

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhukovskii, M.E., Weak Zero-One Law for Random Distance Graphs, Vestn. Ross. Univ. Druzhby Narodov, Ser. Mat. Inform. Fiz., 2010, no. 2 (1), pp. 11–25.

  12. Zhukovskii, M.E., Weak Zero-One Laws for Random Distance Graphs, Dokl. Ross. Akad. Nauk, 2010, vol. 430, no. 3, pp. 314–317 [Dokl. Math. (Engl. Transl.), 2010, vol. 81, no. 1, pp. 51–54].

    MathSciNet  Google Scholar 

  13. Zhukovskii, M.E., The Weak Zero-One Law for Random Distance Graphs, Teor. Veroyatn. Primen., 2010, vol. 55, no. 2, pp. 344–349 [Theory Probab. Appl. (Engl. Transl.), 2010, vol. 55, no. 2, pp. 356–360].

    MathSciNet  Google Scholar 

  14. Raigorodskii, A.M., The Borsuk Problem and the Chromatic Numbers of Some Metric Spaces, Uspekhi Mat. Nauk, 2001, vol. 56, no. 1, pp. 107–146 [Russian Math. Surveys (Engl. Transl.), 2001, vol. 56, no. 1, pp. 103–139].

    MathSciNet  Google Scholar 

  15. Raigorodskii, A.M., Lineino-algebraicheskii metod v kombinatorike (Linear-Algebraic Method in Combinatorics), Moscow: MCCME, 2007.

    Google Scholar 

  16. Mikhailov, K.A. and Raigorodskii, A.M., On the Ramsey Numbers for Complete Distance Graphs with Vertices in {0, 1}n, Mat. Sb., 2009, vol. 200, no. 12, pp. 63–80.

    MathSciNet  Google Scholar 

  17. MacWilliams, F.J. and Sloane, N.J.A. The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz’, 1979.

    MATH  Google Scholar 

  18. Ehrenfeucht, A., An Application of Games to the Completeness Problem for Formalized Theories, Fund. Math., 1960/1961, vol. 49, pp. 121–149.

    MathSciNet  Google Scholar 

  19. Shwentick, T., On Winning Ehrenfeucht Games and Monadic NP, Ann. Pure Appl. Logic, 1996, vol. 79, no. 1–2, pp. 61–92.

    Article  MathSciNet  Google Scholar 

  20. Brenner, J. and Cummings, L., The Hadamard Maximum Determinant Problem, Amer. Math. Monthly, 1972, vol. 79, no. 6, pp. 626–630.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Probability Theory Chair, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    M. E. Zhukovskii

Authors
  1. M. E. Zhukovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. E. Zhukovskii.

Additional information

Original Russian Text © M.E. Zhukovskii, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 3, pp. 39–58.

Supported in part by the Russian Foundation for Basic Research, project no. 09-01-00294, and the President of the Russian Federation Grant for State Support of Young Russian Scientists-Doctors of Sciences, grant no. MD-8390.2010.1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhukovskii, M.E. On a sequence of random distance graphs subject to the zero-one law. Probl Inf Transm 47, 251–268 (2011). https://doi.org/10.1134/S0032946011030045

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0032946011030045

Keywords

Search

Navigation