Large Sets Of Pure Oriented Triple Systems

There are two kinds of oriented triples on X:cyclic triple and transitive triple. A cyclic triple on X is a set of three ordered pairs (x, y), (y,z) and (z,x) of X, which is denoted by (or , or ), and a transitive triple on X is a set of three ordered pairs (x,y), (y,z) and (x,z) of X, which is denoted by (x,y,z).An oriented triple system of order v with indexλis a pair (X, B) where X is a v-set and B is a collection of oriented triples on X, called blocks, such that every ordered pair of X belongs to exactlyλblocks of B. If B consists of cyclic (or transitive) triples only, the system is called a Mendelsohn triple system (or directed triple system) of order v with indexλand denoted by MTS (v,λ) (or DTS (v,λ)). If B contains both cyclic triples and transitive triples, the system is called a hybrid triple system of order v with indexλand denoted by HTS (v,λ).A triple system is called simple if there are no repeated blocks in B. A simple MTS (v,λ) is called pure and denoted by PMTS (v,λ) if ∈B implies (?) B. A simple DTS(v,λ) is called pure and denoted by PDTS(v,λ) if (x,y,z)∈B im-plies (z, y, x), (z, x, y), (y, x, z), (y, z, x), (x, z, y)(?)B. A simple HTS (v,λ) is called pure and denoted by PHTS(v,λ) if one element of the block set{(x,y,z), (z,y,x), (z,x,y), (y, x, z), (y, z, x), (x, z, y), , } is contained in B then the others will not be contained in B.A large set of disjoint MTS(v,λ)s (or DTS(v,λ)s), denoted by LMTS(v,λ) (or LDTS (v,λ)), is a collection of{(X, Bi)}i where each (X, Bi) is an MTS (v,λ) (or DTS (v,λ)) and UiBi is a partition of all cyclic (or transitive) triples on X. A large set of disjoint HTS (v,λ)s, denoted by LHTS (v,λ), is a collection of{(X, Bi)}i where each (X, Bi) is an HTS (v,λ) and UiBi is a partition of all cyclic and transitive triples on X. Furthermore if each (X, Bi) is a PMTS (v,λ) (or PDTS (v,λ), PHTS (v,λ)), then the large set is denoted by LPMTS(v,λ) (or LPDTS(v,λ), LPHTS(v,λ)). In 1850 Kirkman posed a simple problem in Lady's and Gentleman's Diary as follows:Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast.In modern language, we are to find out a Kirkman triple system of order 15 (KTS (15)).Early on in the study of the Kirkman schoolgirls problem, an interesting extension was suggested by Sylvester:Can the 455 triples of the elements of a 15-set be grouped into 13 different KTS (15) designs?This is the first large set problem in the Mathematics history, which excited great interest among amateurs and professionals. From then on various large set problems were posed and investigated. As far as large sets of oriented triple systems are con-cerned, the existence problems of LMTS (v,λ), LDTS (v,λ), LHTS (v,λ), LPMTS (v,1) and LPDTS(v,1) have been completely settled till now. However, there is a little re-search on LPMTS (v,λ), LPDTS (v,λ) or LPHTS (v,λ).In this thesis we investigate such large set problems as LPMTS (v,λ), LPDTS (v,λ) and LPHTS (v,λ).The main results of this thesis are summarized below:(1) There exists an LPDTS (v,2) if and only if v≡0,4 (mod 6), v≥4.(2) There exists an LPMTS (v,2) if and only if v≡0,4 (mod 6), v≥6.(3) For any v≡8,14 (mod 18), v≠14, there exists an LPMTS(v,3).(4) There exists an LPHTS (v,2) for v≡0,4 (mod 6), v≥4 except two infinite families:v(?)6,22 (mod 24), v> 6.(5) Let u, v, A, n be positive integers, u≡1,5 (mod 6), v> 3, v≡0 (mod A), and v>3λ. There exist an LPDTS (uv+2,λ) and an LPHTS (uv+2,λ) where v satisfies one of the following conditions:(ⅰ) v≡2,10 (mod 12);(ⅱ) v＝2n,n≥1, n≠4; (ⅲ) v=3mw-1, m≥0, w≡3,11 (mod 12), w≡9,41 (mod 48), or w=2n+1, n≥1, n≠4.(6) Let u, v,λ, n be positive integers, u≡1,5 (mod 6). There exists an LPMTS (uv+ 2, A) for the following orders v if v>3λand v≡0 (mod A):(ⅰ) v≡2,10 (mod 12) or v≡8 (mod 24);(ⅱ) v=2n, n>2, n≠4;(ⅲ) v=3mw-1, m≥0, w≡3,11 (mod 12), w≡9,41 (mod 48), or w=2n+1, n≥1,n≠2,4.The thesis is divided into five chapters.Chapter 1 This is an introductory chapter. The historical backgrounds of large set problems are recalled and the recent developments on the related problems are in-troduced. Our main methods and results are also listed.Chapter 2 In this chapter we obtain some results about the existence for LPDTS (v,λ) and determine the spectrum of LPDTS (v,2). Firstly a product construction for LPDTS(v,λ) is displayed. Secondly we introduce the definition of LPDTS*(v) and show the relationship of LPDTS (v, A) and LPDTS*(v), then we establish recursive con-structions for LPDTS*(v). Thirdly more constructions of LPDTS*(v) via good large sets are obtained, and we get infinite families for the existence of LPDTS (v,λ). Finally the existence spectrum of LPDTS (v,2) is determined.Chapter 3 In this chapter we obtain some infinite families for the existence of LPMTS (v,λ) and determine the spectrum of LPMTS (v,2). A product construction for LPMTS (v,λ) is established at the beginning of this chapter. Then we introduce the definition of LPMTS*(v) and show the relationship of LPMTS (v,λ) and LPMTS*(v). Combining the known results of LPMTS*(v), we get infinite families for the existence of LPMTS (v,λ). Then the spectrum of LPMTS (v,2) is determined. Meanwhile an infinite family for the existence of LPMTS (v,3) is obtained.Chapter 4 In this chapter new infinite families for the existence of LPHTS (v,λ) are displayed. Especially we point out that except two infinite families the necessary condition for the existence spectrum of LPHTS (v,2) is also sufficient. Chapter 5 This is the last chapter of the thesis. In this chapter several further research problems on large sets are presented.