On Defining Sets of Full Designs with Block Size Three

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 dD of D there exists a minimal defining set dF of F such that dD = dF ∩D. The unique simple design with parameters ( v, k, ( v − 2 k − 2 )) ∗Donovan and Lefevre supported by grants DP0664030 and LX0453416 †This work was carried out at The University of Queensland and Yazıcı was supported by Raybould fellowship and TUBITAK CAREER grant 106T574 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.