Skip to main content
SpringerLink
Log in

Indecomposable large sets of Steiner triple systems with indices 5, 6

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

A family (X, B 1), (X, B 2), …, (X, B q ) of q STS(υ)s is a λ-fold large set of STS(υ) and denoted by LSTS λ (υ) if every 3-subset of X is contained in exactly λ STS(υ)s of the collection. It is indecomposable and denoted by IDLSTS λ (υ) if there does not exist an LSTSλ′(υ) contained in the collection for any λ′ < λ. In this paper, we show that for λ = 5, 6, there is an IDLSTS λ (υ) for υ ≡ 1 or 3 (mod 6) with the exception IDLSTS6(7).

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. Kirkman, T. P.: On a problem in combinations. Cambridge and Dublin Math. J., 2, 191–204 (1847)

    Google Scholar 

  2. Cayley, A.: On the triadic arrangements of seven and fifteen things. London Edinburgh and Dublin Philos. Mag. and J. Sci., 37(3), 50–53 (1850) (Collected Mathematical papers I, 481–484)

    Google Scholar 

  3. Lu, J. X.: On large sets of disjoint Steiner triple systems I, II, and III. J. Combin. Theory (A), 34, 140–146, 147–155 and 156–182 (1983)

    Article  Google Scholar 

  4. Lu, J. X.: On large sets of disjoint Steiner triple systems IV, V, and VI. J. Combin. Theory (A), 37, 136–163, 164–188, and 189–192 (1984)

    Article  MATH  Google Scholar 

  5. Teirlinck, L.: A completion of Lu’s determination of the spectrum of large sets of disjoint Steiner triple systems. J. Combin. Theory (A), 57, 302–305 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ji, L. J.: A new existence proof for large sets of disjoint Steiner triple systems. J. Combin. Theory (A), 112, 308–327 (2005)

    Article  MATH  Google Scholar 

  7. Griggs, T. S., Rosa, A.: Large sets of Steiner triple systems in “Surveys in Combinatorics” (ed. P. Rowlinson), Vol. 218, London Math. Soc., Lecture Note Series, Cambridge University Press, 1995, 25–39

  8. Ji, L. J., Tian, Z. H., Kang, Q. D.: λ-fold indecomposable large sets of Steiner triple systems. Sci. China, Ser. A, 53, 2877–2888 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mohácsy, H., Ray-Chaudhuri, D. K.: Candelabra systems and designs. J. Statist. Plann. Inference, 106, 419–448 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hartman, A., Phelps, K. T.: Steiner quadruple systems. In: Contemporary Design Theory, Wiley, New York, 1992, 205–240

    Google Scholar 

  11. Lenz, H.: Tripling quadruple quadruples. Ars Combin., 20, 193–202 (1985)

    MathSciNet  MATH  Google Scholar 

  12. Cao, H. T., Ji, L. J., Zhu, L.: Large sets of disjoint packings on 6k + 5 points. J. Combin. Theory (A), 108, 169–183 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rosa, A.: A theorem on the maximum number of disjoint Steiner triple systems. J. Combin. Theory (A), 18, 305–312 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Teirlinck, L.: On the maximum number of disjoint triple systems. Discrete Math., 1, 299–300 (1973)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Hebei Normal University, Shijiazhuang, 050016, P. R. China

    Mei Hui Cheng & Zi Hong Tian

Authors
  1. Mei Hui Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zi Hong Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zi Hong Tian.

Additional information

Supported by National Natural Science Foundation of China (Grant Nos. 10971051 and 11071056)

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cheng, M.H., Tian, Z.H. Indecomposable large sets of Steiner triple systems with indices 5, 6. Acta. Math. Sin.-English Ser. 28, 2169–2184 (2012). https://doi.org/10.1007/s10114-012-0693-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-012-0693-z

Keywords

MR(2000) Subject Classification

Search

Navigation