On defining sets of full designs with block size three

Abstract

A defining set of a t-(v, k, lambda) 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 {vertical bar M vertical bar vertical bar M is a minimal defining set of D}. We show that if a t-(v, k, lambda) 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) boolean and D. The unique simple design with parameters (v, k, ((v-2)(k-2))) 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, lambda) 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, lambda) 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.