Large Sets Problems On Combinatorial Designs

A block design is triple D=(X,B,I),where X is a finite set(the point set)of order v,B is the block set and I is the incidence relation between X and B.For the given block design D,if all the blocks from X,which have the same type of the blocks in B,can be partitioned into some Bi,such that each(X,Bi,I)is a block design with the same parameters and the same type as D,the {(X,Bi,I)}i is called as large set of D-design of order v.In this thesis,we focus on some types of large sets which are LKTS,FDGDD(3,4,v{2}),LS(gns1)and LS+(2n41).The thesis is divided into five chapters.The first chapter is an introduction,some historical backgrounds,definitions of the large set problems are recalled and our main methods and results are also listed.The second chapter focus on the direct constructions of LKTS(v),where v ? 400.Suppose v = q+ 2,q is a prime power,we almost establish the existence of LKTS(v)by considering the two cases of q?19(mod 24),q ? 1,13(mod 24).The third chapter discuss the existence of FDGDD(3,4,v{2}),we proved that there exist FDGDD(3,4,v{2}),v? {23,29,47}.Further more,we can establish a relation between the the asymptotic existence of FDGDD(3,4,v{2})and LKTS(18n + 9).The fourth chapter gives a complete solution of the existence LS(2n41)by con-structing five remaining values.Based on the existence of LS(2n41),LS+(2n41)s is put forward and we give a nearly complete solution of its existence?The fifth chapter give some unsettled problems in this thesis.Our main results is as follows:(1)There exists a LKTS(q + 2)for any prime power q<400 and q? 1(mod 6),possibly except that q ? {373,397}.(2)There exists a FDGDD(3,4,v{2}),v ?{23,29,47}.(3)There exists a LS(2n41)if and only if n ? 0(mod 3).(4)There exists a LS+(2n41)if and only if n ? 0(mod 3)with three possible exceptions n ? {30,48,144}.