Skip to main content
SpringerLink
Log in

Spectra of first-order formulas with a low quantifier depth and a small number of quantifier alternations

  • Mathematics
  • Published:
Doklady Mathematics Aims and scope Submit manuscript

Abstract

Spectra of first-order formulas are studied. The spectrum of a first-order formula is the set of all positive α such that either this formula is true for the random graph G(n, n −α) with an asymptotic probability being neither 0 nor 1 or the limit does not exist. It is well known that there exists a first-order formula with an infinite spectrum. The minimum number of quantifier alternations in such a formula is found.

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. N. K. Vereshchagin and A. Shen’, Languages and Calculi (MNTsMO, Moscow, 2000) [in Russian].

    Google Scholar 

  2. S. Shelah and J. H. Spencer, J. Am. Math. Soc. 1, 97–115 (1988).

    Article  Google Scholar 

  3. Yu. V. Glebskii, D. I. Kogan, M. I. Liagonkii, and V. A. Talanov, Kibernetika 2, 17–26 (1969).

    Google Scholar 

  4. R. Fagin, J. Symbol. Logic 41, 50–58 (1976).

    Article  Google Scholar 

  5. M. E. Zhukovskii, Discrete Math. 312, 1670–1688 (2012).

    Article  MathSciNet  Google Scholar 

  6. M. E. Zhukovskii, Eur. J. Combin. 60, 66–81 (2017).

    Article  Google Scholar 

  7. M. E. Zhukovskii and L. B. Ostrovskii, Dokl. Math. 94 (2), 555–557 (2016).

    Article  Google Scholar 

  8. M. E. Zhukovskii and A. M. Raigorodskii, Russ. Math. Surv. 70 (1), 33–81 (2015).

    Article  Google Scholar 

  9. J. H. Spencer, Combinatorica 10 (1), 95–102 (1990).

    Article  MathSciNet  Google Scholar 

  10. M. Zhukovskii, Moscow J. Combin. Number Theory 6 (4), 73–102 (2016).

    Google Scholar 

  11. B. Bollobás, Math. Proc. Cambridge Phil. Soc. 90, 197–206 (1981).

    Article  Google Scholar 

  12. A. Ruciński and A. Vince, Congr. Numerantim 49, 181–190 (1985).

    Google Scholar 

  13. J. H. Spencer, J. Combin. Theory Ser. A 53, 286–305 (1990).

    Article  MathSciNet  Google Scholar 

  14. N. Alon and J. H. Spencer, Probabilistic Method, 3rd ed. (Wiley, New York, 2008).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tambov State University, Tambov, Russia

    M. E. Zhukovskii

  2. Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow oblast, 141700, Russia

    A. D. Matushkin

Authors
  1. M. E. Zhukovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. D. Matushkin
    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, A.D. Matushkin, 2017, published in Doklady Akademii Nauk, 2017, Vol. 475, No. 2, pp. 127–129.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhukovskii, M.E., Matushkin, A.D. Spectra of first-order formulas with a low quantifier depth and a small number of quantifier alternations. Dokl. Math. 96, 326–328 (2017). https://doi.org/10.1134/S1064562417040093

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562417040093

Search

Navigation