A completion of the spectrum for large sets of disjoint transitive triple systems

In this paper, we give a recursive construction from an LTTS(v + 2) to an LTTS(16v + 2) for v 3. Furthermore, the existence of LTTS(2ⁿ + 2) is proved. Thereby, we completely solve the existence problem of LTTS)(v) (large set of pairwise disjoint transitive triple systems of order v).     

On large sets of disjoint Kirkman triple systems

In this paper, we introduce LR(u) designs and use these designs together with large sets of Kirkman triple systems (LKTS) and transitive KTS (TKTS) of order v to construct an LKTS(uv). Our main result is that there exists an LKTS(v) for v∈{3nm(2·13k+1)t;n?1,k?1,t=0,1,m∈{1,5,11,17,25,35,43}}.     

A construction for large sets of disjoint Kirkman triple systems

A Steiner system S(t, k, v) is called i-resolvable, 0 < i < t, if its block set can be partitioned into S(i, k, v). In this paper, a 2-resolvable S(3, 4, v) is used to construct a large set of disjoint Kirkman triple systems of order 3v − 3 (briefly LKTS) and some new orders for LKTS are then obtained. Research supported by Tianyuan Mathematics Foundation of NSFC Grant 10526032 and Natural Science Foundation of Universities of Jiangsu Province Grant 05KJB110111.     

On large sets of disjoint Steiner triple systems II

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, we prove that if n is an odd number, there exist 12 mutually orthogonal Latin squares of order n and D(1 + 2n) = 2n ? 1, then D(1 + 12n) = 12n ? 1.     

On large sets of disjoint steiner triple systems I

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, it is proved that if D(2 + n) = n, p is a prime number, p ≡ 7 (mod 8) or p? {5, 17, 19, 2}, and (p, n) ≠ (5, 1), then D(2 + pn) = pn.     

On large sets of disjoint steiner triple systems,VI

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, we prove that D(v) = v ? 2 holds for all v ≡ 1, 3 (mod 6) (v>7), except possibly v = 141, 283, 501, 789, 1501, 2365.     

The spectrum for large sets of pure directed triple systems

An LPDTS(ν) is a collection of 3(ν-2) disjoint pure directed triple systems on the same set ofνelements. It is showed in Tian's doctoral thesis that there exists an LPDTS(ν) forν=0,4 (mod 6),ν≥4. In this paper, we establish the existence of an LPDTS(ν) forν= 1,3 (mod 6),ν> 3. Thus the spectrum for LPDTS(ν) is completely determined to be the set {ν:ν= 0, 1 (mod 3),ν≥4}.     

The spectrum for large sets of pure Mendelsohn triple systems

An LPMTS(v) is a collection of v-2 disjoint pure Mendelsohn triple systems on the same set of v elements. In this paper, the concept of t-purely partitionable Mendelsohn candelabra system (or t-PPMCS in short) is introduced for constructing LPMTS(v)s. A powerful recursive construction for t-PPMCSs is also displayed by utilizing s-fan designs. Together with direct constructions, the existence of an LPMTS(v) for and v>1 is established. For odd integer v?7, a special construction from both LPMTS(v) and OLPMTS(v) to LPMTS(2v+1) is set up. Finally, the existence of an LPMTS(v) is completely determined to be the set .     

On large sets of disjoint steiner triple systems,V

In a previous paper (J. Combin. Theory Ser. A34 (1983), 156–182), to construct large sets of disjoint STS(3n)'s (i.e., LTS(3n)'s), a kind of combinatorial design, denoted by LD(n), where n is the order of design, was introduced and it was shown that if there exist both an LD(n) and an LTS(n + 2), then there exists an LTS(3n) also. In this paper, after having established some recursive theorems of LD(n), the following result was proved: If n is a positive integer such that n≡11 (mod 12), then there exists an LD(n), except possibly n ∈ {23, 47, 59, 83, 107, 167, 179, 227, 263, 299, 347, 383, 719, 767, 923, 1439}.     

A new existence proof for large sets of disjoint Steiner triple systems

A Steiner triple system of order v (briefly STS(v)) consists of a v-element set X and a collection of 3-element subsets of X, called blocks, such that every pair of distinct points in X is contained in a unique block. A large set of disjoint STS(v) (briefly LSTS(v)) is a partition of all 3-subsets (triples) of X into v-2 STS(v). In 1983–1984, Lu Jiaxi first proved that there exists an LSTS(v) for any v≡1 or with six possible exceptions and a definite exception v=7. In 1989, Teirlinck solved the existence of LSTS(v) for the remaining six orders. Since their proof is very complicated, it is much desired to find a simple proof. For this purpose, we give a new proof which is mainly based on the 3-wise balanced designs and partitionable candelabra systems.     

Constructions for overlarge sets of disjoint pure directed triple systems

Let X be a v-set, v≥3. A transitive triple (x,y,z) on X is a set of three ordered pairs (x,y),(y,z) and (x,z) of X. A directed triple system of order v, denoted by DTS(v), is a pair (X,?), where X is a v-set and ? is a collection of transitive triples on X such that every ordered pair of X belongs to exactly one triple of ?. A DTS(v) is called pure and denoted by PDTS(v) if (x,y,z)∈? implies (z,y,x)??. An overlarge set of disjoint PDTS(v) is denoted by OLPDTS(v). In this paper, we establish some recursive constructions for OLPDTS(v), so we obtain some results.     

The spectrum for overlarge sets of directed triple systems

A directed triple system of order v,denoted by DTS(v,λ),is a pair(X,B)where X is a v- set and B is a collection of transitive triples on X such that every ordered pair of X belongs toλtriples of B.An overlarge set of disjoint DTS(v,λ),denoted by OLDTS(v,λ),is a collection{(Y\{y},A_i)}_i, such that Y is a(v 1)-set,each(Y\{y},A_i)is a DTS(v,λ)and all A_i's form a partition of all transitive triples of Y.In this paper,we shall discuss the existence problem of OLDTS(v,λ)and give the following conclusion:there exists an OLDTS(v,λ)if and only if eitherλ=1 and v≡0,1(mod 3),orλ=3 and v≠2.     

The spectrum for overlarge sets of hybrid triple systems

A hybrid triple system of order v and index λ,denoted by HTS(v,λ),is a pair(X,B) where X is a v-set and B is a collection of cyclic triples and transitive triples on X,such that every ordered pair of X belongs to λ triples of B. An overlarge set of disjoint HTS(v,λ),denoted by OLHTS(v,λ),is a collection {(Y {y},Ai)}i,such that Y is a(v+1)-set,each(Y {y},Ai) is an HTS(v,λ) and all Ais form a partition of all cyclic triples and transitive triples on Y.In this paper,we shall discuss the existence problem of OLHTS(v,λ) and give the following conclusion: there exists an OLHTS(v,λ) if and only if λ=1,2,4,v ≡ 0,1(mod 3) and v≥4.     

On sets of pairwise disjoint blocks in Steiner triple systems

In a Steiner triple system with 19 points, each disjoint pair blocks is contained in at least 43 quadruplets of pairwise disjoint blocks. In a Steiner triple system with 25 points, each disjoint pair of blocks is contained in a pairwise disjoint quintuple of blocks. Theorems used are those of Connor on determinants based on intersecting and nonintersecting blocks of a BIBD, and of Turán on extremal graphs without triangles.     

Embeddings of resolvable mendelsohn triple systems

In this article it is shown that any resolvable Mendelsohn triple system of order u can be embedded in a resolvable Mendelsohn triple system of order v iff v≥ 3u, except possibly for 71 values of (u,v). © 1993 John Wiley & Sons, Inc.     

Embeddings of pure mendelsohn triple systems and pure directed triple systems

It is proved in this article that the necessary and sufficient conditions for the embedding of a λ-fold pure Mendelsohn triple system of order v in λ-fold pure Mendelsohn triple of order u are λu(u ? 1) ≡ 0 (mod 3) and u ? 2v + 1. Similar results for the embeddings of pure directed triple systems are also obtained. © 1995 John Wiley & Sons, Inc.     

The existence spectrum for overlarge sets of pure directed triple systems

A directed triple system of order v,denoted by DTS(v),is a pair (X,B) where X is a v-set and B is a collection of transitive triples on X such that every ordered pair of X belongs to exactly one triple of B.A DTS(v) (X,A) is called pure and denoted by PDTS(v) if (a,b,c) ∈ A implies (c,b,a) ∈/ A.An overlarge set of PDTS(v),denoted by OLPDTS(v),is a collection {(Y \{yi},Aij) : yi ∈ Y,j ∈ Z3},where Y is a (v+1)-set,each (Y \{yi},Aij) is a PDTS(v) and these Ais form a partition of all transitive triples on Y .In this paper,we shall discuss the existence problem of OLPDTS(v) and give the following conclusion: there exists an OLPDTS(v) if and only if v ≡ 0,1 (mod 3) and v 3.     

On the embeddings of mendelsohn triple systems

It is proved in this paper that there exists an incomplete Mendelsohn triple system IMTS(u,v; λ) if and only ifλ(u-v)(u-2v-1)≡0(mod 3),u≥2v+1 and (u, v, λ) ≠ (6, 1, 1). As a consequence, it is proved that for any given λ≥1, a Mendelsohn triple system MTS (v, λ) can be embedded in an MTS (u, λ) if and only ifλu(u-1)≡0(mod 3) andu≥2v+1. Project supported by the National Natural Science Foundation of China.