Another Look at Large Sets of Steiner Triple Systems

  • Martin J. Sharry
  • Anne Penfold Street

Abstract

If v = 1 or 3 mod 6 and v > 7, then there exists a large set of Steiner triple systems of order v, LS (STS (v)).

This result was largely known by 1984, though the six cases which were unsettled then have since been solved, and the proof simplified. The proof is by construction, using a fairly small number of initial large sets. Each construction is explained, often with the aid of tables of blocks. In the interests of brevity, proofs and complete examples are not given, though the ingredients needed for the construction of the examples are. The recursive constructions are of three types:
  1. (i)

    extension of a large set of STS (v) to a large set of STS (3v) by using idempotent commutative quasigroups;

     
  2. (ii)

    extension of a large set of STS (v)to a large set of STS (2v + 1) by using good one-factorizations;

     
  3. (iii)

    special constructions to fill in the remaining cases.

     

Keywords

Triple System Counting Argument Steiner Triple System Balance Incomplete Block Design Transversal Design 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. C. Bose, On the construction of balanced incomplete block designs, Ann. Eugenics 9 (1939), 353–399.MathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Cayley, On the triadic arrangements of seven and fifteen things, Lon-don, Edinburgh & Dublin Phil. Mag. J. Sci. 37 (1850), 50–53Google Scholar
  3. [3]
    R. H. F. Denniston, Some packings with Steiner triple systems, Discrete Math. 9 (1973), 213–227.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Jean Doyen, Constructions of disjoint Steiner triple systems, Proc. Amer. Math. Soc. 32 (1972), 409–416.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Haim Hanani, On quadruple systems, Canadian J. Math. 12 (1960), 145–157.MATHGoogle Scholar
  6. [6]
    Haim Hanani, D. K. Ray-Chaudhuri and Richard M. Wilson, On resolvable designs, Discrete Math. 3 (1972), 343–357.MATHGoogle Scholar
  7. [7]
    Thomas P. Kirkman, On a problem in combinations, Cambridge and Dublin Math. J. 2 (1847), 191–204.Google Scholar
  8. [8]
    Lu Jia-Xi, On large sets of disjoint Steiner triple systems I, J. Combinatorial Theory 34A (1983), 140–146.CrossRefGoogle Scholar
  9. [9]
    Lu Jia-Xi, On large sets of disjoint Steinner triple systems II, J. Combinatorial Theory 34A (1983), 147–155.CrossRefGoogle Scholar
  10. [10]
    Lu Jia-Xi, On large sets of disjoint Steiner triple systems III, J. Combinatorial Theory 34A (1983), 156–182.CrossRefGoogle Scholar
  11. [11]
    Lu Jia-Xi, On large sets of disjoint Steiner triple systems IV, J. Combinatorial Theory 37A (1984), 136–163.CrossRefGoogle Scholar
  12. [12]
    Lu Jia-Xi, On large sets of disjoint Steiner triple systems V, J. Combinatorial Theory 37A (1984), 164–188.CrossRefGoogle Scholar
  13. [13]
    Lu Jia-Xi, On large sets of disjoint Steiner triple systems VI, J. Combinatorial Theory 37A (1984), 189–192.CrossRefMATHGoogle Scholar
  14. [14]
    Lu Jia-Xi, On large sets of disjoint Steiner triple systems VII, unfinished manuscript.Google Scholar
  15. [15]
    Eric Mendelsohn and Alexander Rosa, One-factorizations of the complete graph — a survey, J. Graph Theory 9 (1985), 43–65.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    D. K. Ray-Chaudhuri and Richard M. Wilson, Solution of Kirkman’s school girl problem, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, Rhode Island, 19 (1971), 187–203.MathSciNetGoogle Scholar
  17. [17]
    Alexander Rosa, A theorem on the maximum number of disjoint Steiner triple systems, J. Combinatorial Theory 18A (1975), 305–312.CrossRefMATHGoogle Scholar
  18. [18]
    Martin J. Sharry, Large and overlarge sets of block designs, Ph. D. thesis, The University of Queensland, Queensland, Australia, 1992.Google Scholar
  19. [19]
    D. R. Stinson, A short proof of the non—existence of a pair of orthogonal Latin squares of order six, J. Combinatorial Theory 36A (1984), 373–376.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Anne Penfold Street and Deborah J. Street, Combinatorics of Experimental Design, Clarendon Press, Oxford, New York (1987).Google Scholar
  21. [21]
    Luc Teirlinck, On the maximum number of disjoint Steiner triple systems, Discrete Math. 6 (1973), 299–300.MathSciNetMATHGoogle Scholar
  22. [22]
    Luc Teirlinck, A completion of Lu’s determination of the spectrum for large sets of disjoint Steiner triple systems, J. Combinatorial Theory 57A (1991), 302–305.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Martin J. Sharry
    • 1
  • Anne Penfold Street
    • 1
  1. 1.Centre for Combinatorics, Department of MathematicsThe University of QueenslandQueenslandAustralia

Personalised recommendations