On The Spectrum of Minimal Defining Sets of Full Designs

A defining set of a t-(v,k,λ ) design is a subcollection of the block set of the design which is not contained in any other design with the same parameters. A defining set is said to be minimal if none of its proper subcollections is a defining set. A defining set is said to be smallest if no other defining set has a smaller cardinality. A t-(v,k,λ ) design D = (V,B) is called a full design if B is the collection of all possible k-subsets of V . Every simple t-design is contained in a full design and the intersection of a defining set of a full design with a simple t-design contained in it, gives a defining set of the corresponding t-design. With this motivation, in this paper, we study the full designs when t = 2 and k = 3 and we give several families of non-isomorphic minimal defining sets of full designs. Also, it is proven that there exist values in the spectrum of the full design on v elements such that the number of non-isomorphic minimal defining sets on each of these sizes goes to infinity as v→ ∞. Moreover, the lower bound on the size of the defining sets of the full designs is improved by finding the size of the smallest defining sets of the full designs on 8 and 9 points. Also, all smallest defining sets of the full designs on 8 and 9 points are classified.