Abstract
A defining set of a t-(v, k, λ) design is a subcollection of its blocks which is contained in no other t-design with the given parameters, on the same point set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M| | M is a minimal defining set of D}. We show that if a t-(v, k, λ) design D is contained in a design F, then for every minimal defining set d D of D there exists a minimal defining set d F of F such that \({d_D = d_F\cap D}\). The unique simple design with parameters \({{\left(v,k, {v-2\choose k-2}\right)}}\) is said to be the full design on v elements; it comprises all possible k-tuples on a v set. Every simple t-(v, k, λ) design is contained in a full design, so studying minimal defining sets of full designs gives valuable information about the minimal defining sets of all t-(v, k, λ) designs. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. Several families of non-isomorphic minimal defining sets of these designs are found. For given v, a lower bound on the size of the smallest and an upper bound on the size of the largest minimal defining set are given. The existence of a continuous section of the spectrum comprising approximately v values is shown, where just two values were known previously.
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Akbari S., Maimani H.R., Maysoori Ch.: Minimal defining sets for full 2 − (v, 3, v − 2) designs. Australas. J. Combin. 23, 5–8 (2001)
Gower R.A.H.: Minimal defining sets in a family of Steiner triple systems. Australas. J. Combin. 8, 55–73 (1993)
Gray B.D., Mathon R., Moran T., Street A.P.: The spectrum of minimal defining sets of some Steiner systems. Discrete Math. 261, 277–284 (2003)
Gray K.: On the minimum number of blocks defining a design. Bull. Austral. Math. Soc. 41, 97–112 (1990)
Gray K.: Further results on smallest defining set of well known designs. Australas. J. Combin. 1, 91–100 (1990)
Gray K.: Defining sets of single-transposition-free designs. Util. Math. 38, 97–103 (1990)
Gray K., Street A.P.: On defining sets of full designs and of designs related to them. J. Combin. Math. Combin. Comput. 60, 97–104 (2007)
Gray K., Street A.P.: Constructing defining sets of full designs. Util. Math. 76, 91–99 (2008)
Gray K., Street A.P.: Defining Sets, Section IV.13 In: Colbourn C.J., Dinitz J.H. (eds). CRC Handbook of Combinatorial Designs, 2nd edn, pp. 382–385. CRC Press, Boca Raton (2007)
Havas G., Lawrence J.L., Ramsay C., Street A.P., Yazıcı E.Ş.: Defining set spectra for designs can have arbitrarily large gaps. Util. Math. 75, 67–81 (2008)
Kolotoğlu, E.: A new algorithm for finding the complete list of minimal defining sets of t-designs, Masters Thesis, Koç University, July 2007
Yazıcı, E.Ş., Kolotoğlu, E.: On minimal defining sets of full designs and self-complementary designs, and a new algorithm for finding defining sets of t-designs, Graphs Combin. (2010) (to appear)
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D. Donovan and J. Lefevre supported by grants DP0664030 and LX0453416.
Yazıcı was supported by Raybould fellowship and TUBITAK CAREER grant 106T574.
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Donovan, D., Lefevre, J., Waterhouse, M. et al. On Defining Sets of Full Designs with Block Size Three. Graphs and Combinatorics 25, 825–839 (2009). https://doi.org/10.1007/s00373-010-0882-4
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DOI: https://doi.org/10.1007/s00373-010-0882-4