Abstract
Let \(J^{3}_{R}(v)\) denote the set of all integers k such that there exists a collection of three KTS(v) pairwise intersecting in the same set of k blocks. In this paper for sufficiently large v where \(v\equiv 3\) (mod 6), we determine the set \(J^{3}_{R}(v)\) except possibly for two values \(t_v-10\) and \(t_v-13\), where \(t_v\) is the number of triples contained in KTS(v).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brouwer, A.E.: Optimal packings of \(k_4\)’s into a \(k_n\). J. Comb. Theory Series A 26(3), 278–297 (1979)
Chang, Y., Lo Faro, G.: Intersection numbers of Kirkman triple systems. J. Comb. Theory Series A 86(2), 348–361 (1999)
Colbourn, C.J., Dinitz, J.H.: Handbook of combinatorial designs. CRC Press (2007)
Hanani, H.: Balanced incomplete block designs and related designs. Discrete Math. 11(3), 255–369 (1975)
Lindner, C.C., Rodger, C.A.: Design theory. CRC Press (2008)
Lindner, C.C., Rosa, A.: Steiner triple systems having a prescribed number of triples in common. Can. J. Math. 27(5), 1166–1175 (1975)
Milici, S., Quattrocchi, G.: On the intersection problem for three Steiner triple systems. Ars Comb. A 24, 174–194 (1987)
Rashidi, S., Soltankhah, N.: On the possible volume of three way trades. Electron. Notes Discrete Math. 43, 5–13 (2013)
Rashidi, S., Soltankhah, N.: The 3-way intersection problem for S(2, 4, v) designs, Utilitas Mathematica (to appear)
Ray-Chaudhuri, D.K., Wilson, R.M.: Solution of Kirkman’s schoolgirl problem. In Proceedings of Symposia in Pure Mathematics, 19, 187–203 (1971)
Rees, R., Stinson, D.R.: On the existence of Kirkman triple-systems containing Kirkman subsystems. Ars Comb. 26, 3–16 (1988)
Shen, H.: Intersections of Kirkman triple systems. J. Stat. Plan. Inference 94(2), 313–325 (2001)
Stinson, D.R.: A survey of Kirkman triple systems and related designs. Discrete Math. 92(13), 371–393 (1991)
Wei, H., Ge, G.: Group divisible designs with block sizes from \(k_{1(3)}\) and Kirkman frames of type \(h^{u}m^{1}\). Discrete Math. 329, 42–68 (2014)
Zhou, J., Chang, Y.: New results on large sets of Kirkman triple systems. Designs Codes Cryptogr. 55(1), 1–7 (2010)
Acknowledgements
The authors of this article would like to acknowledge Professor E. S. Mahmoodian for his interesting suggestions and his useful comments. We are also thankful to Mr. A. Khosroshahi and Mr. M. Hasanvand for helping us to find small cases, with the computer programming. And last but not least we would like to express our appreciation to Ms. S. Golalizadeh and Mr. A. Chegini for their very beneficial discussions and their interest in our work.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Amjadi, H., Soltankhah, N. The 3-Way Intersection Problem for Kirkman Triple Systems. Graphs and Combinatorics 33, 673–687 (2017). https://doi.org/10.1007/s00373-017-1801-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-017-1801-8