Strictly Pairwise Distinct Designs

Let K be a set of positive integers and v be a positive integer.A pairwise distinct design is a pair(X,B),where X is a set of v points and B is a set of subsets of X(called blocks),each of cardinality from K,such that no two pairs of X occur in the same number of blocks.Further,if for each 1≤i≤v(v-1)/2,there exists exactly one pair of X occurring in i blocks,such pairwise distinct design is said to be strict.Stantont present that the necessary conditions for the existence of strictly pairwise distinct triple systems are v＞3 and v≡0,1(mod 3).In this paper the existence of pairwise distinct design are investigated.It is established that the necessary conditions for the existence of a strictly pairwise distinct triple system are also sufficient.In addition,we present that the necessary conditions for the existence of a strictly pairwise distinct group divisible design of type g~t and block size 3 are g≡0(mod 3),or g≡1,2(mod 3)and t≡0,1(mod 3).There are five chapters in this thesis.In Chapter 1,we introduce some basic definitions and some known results of pairwise distinct design.In Chapter 2,we introduce the basic definitions of auxiliary designs,then describe our recursive constructions which are very useful for the proof of the main conclusions in the thesis.In Chapter 3,we present some examples of strictly pairwise distinct triple systems of small order,then present some special new type of block designs.At last,we prove that the necessary conditions for the existence of a strictly pairwise distinct triple systems are also sufficient.In Chapter 4,we present the necessary conditions for the existence of a strictly pairwise distinct group divisible design of type g~t and block size 3,then some examples and basic results of the designs are presented in this section.We also have some examples of strictly pairwise distinct design of block size 4.Finally,we present some examples of strictly pairwise distinct designs with some restrictions.In Chapter 5,the main conclusions of this thesis are summarized,and the further research problems are presented at last.