# On defining sets of full designs with block size three

Donovan, D., Lefevre, J., Waterhouse, M. and Yazici, E. S. (2009) On defining sets of full designs with block size three. Graphs and Combinatorics, 25 6: 825-839. doi:10.1007/s00373-010-0882-4

Author Donovan, D.Lefevre, J.Waterhouse, M.Yazici, E. S. On defining sets of full designs with block size three Graphs and Combinatorics   Check publisher's open access policy 0911-01191435-5914 2009-12 Article (original research) 10.1007/s00373-010-0882-4 25 6 825 839 15 Mikio Kano Japan Springer Japan KK 2010 eng 010104 Combinatorics and Discrete Mathematics (excl. Physical Combinatorics)970101 Expanding Knowledge in the Mathematical SciencesC1 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. Defining setsFull designs C1 Confirmed Code

 Document type: Journal Article Article (original research) School of Mathematics and Physics 2010 Higher Education Research Data Collection

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