On the existence of Kirkman triple systems containing Kirkman subsystems
On the existence of Kirkman triple systems containing Kirkman subsystems
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发表时间：
1988
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通讯作者：
R. Rees;Douglas R Stinson
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作者：
R. Rees;Douglas R Stinson
一对平衡设计（或PBD）是一对（X,,4），使得X是元素的集合（称为“点”），而4是X的子集合（称为“块”），使得每个无序的点对都包含在唯一的“4”块中。如果u是一个正整数，而/(是一个正整数的集合，那么我们说ilrat （X,/t）是a (u， K)PBD，如果lxl = u，对于每个a e a， lal e K，整数u称为PBD的阶数。利用这个符号，我们可以定义一个u阶的Steiner三重系统，我们记作STS(u)为a (u, {3})-PBD。众所周知，当且仅当u = 1或3模5时，STS(u)存在。设（X,"4）为PBD。如果一组点Y g X具有这样的性质：对于任意a e a，要么是n. 41_(1)，要么是a gY，那么我们说Y是PBD的子设计或j1。设计的顺序是l。子设计Y是适当的tfY / x，如果Y是一个suMesign，那么我们可以删除所有的block a C Y，并将ilrem替换为一个单独的block Y，结果是PBD。此外，PBD的任何块或点本身都是一个suMesign。2010年，Doyen和Wilson研究了包含子系统的Steiner文件系统的构造问题。包含STS(u)作为子系统的STS(u)存在的明显必要条件是：u) 2w + L， u: I或3模量6，u: I或3模量6。在[2]中，显示了这些必要条件是充分的：PBD中的并行c/ass是一组块，它们形成了点sel的分区，如果这些块集可以被分区到并行类中，则apbd是可解析的。将阶函数的kirkman3系（orKTS(u)）定义为可解的STS(u)。Ray{haudhuri和Wilson在[14]中证明了当且仅当u:3模时存在KTS(u)。在本文中，我们感兴趣的是包含KTS(u)作为子系统的KTS(u)。只有当KTS(u)的并行类被KTS(u)的并行类诱导时，我们才说KTS（ur）是KTS(u)的子系统。我们将该子系统描述为子kts （ur）。含有子KTS(u)的KTS(u)存在的明显必要条件是u) 3w, v: 1t): 3模6。最近有几篇论文对这个问题进行了研究，并证明了以下结果。
A pairwise balanced design (or, PBD) is a pair (X,,4), such that X is a set of elements (calleA poinrs) and "4 is a set of subses of X (called blocks\, such that every unordered pair of points is contained in a unique block of "4. If u is a positive integer and /( is a set of positive integers, then we say ilrat (X,/t) is a (u, K)PBD if lxl = u, and lal e K for every A e A. The integer u is called the order of the PBD. Using this notation, we can define a Steiner triple system of order u, which we denote STS(u), to be a (u, {3 })-PBD.It is of course wcll-known Lhat an STS(u) exists if and only if u : 1 or 3 modulo 5. Let (X,"4) be a PBD. If a set of points Y g X has the property that, for any A e A, either ly n.4l _( l or A gY , then we say that Y is a subdesign or Jlat ot the PBD. The orderof the suMesign is lYl. The subdesign Y is proper tfY / X. If Y is a suMesign, then we can delete all blocks A C Y and replace ilrem by a single block, Y, and the result is a PBD. Also, any block or point of a PBD is it^self a suMesign. The problem of constructing Steiner fiple systems containing subsystems was studied by Doyen and Wilson in [2]. The obvious necessary conditions for the existence of an STS(u) containing an STS(u) as a subsystem are u ) 2w + L, u: I or3 modulo6,u: I or3 modulo6. In [2],itis shownthatthesenecessary conditions are suffi cienl A parallel c/ass in a PBD is a set of blocks that form a partition of the point sel APBDiS resolvable if theblocksetcanbepartitionedintoparatlelclasses. A Kirkmantriple system of ordcru, orKTS(u), is delined to be aresolvable STS(u). In [14], Ray{haudhuri and Wilson showed rhar there exisrs a KTS(u) if and only if u:3 modulo6. In this paper, we are interested in KTS(u) which contain KTS(u) as subsystems. We say that a KTS(ur) is a subsystem of a KTS(u) only if the parallel classes of the KTS( u) are induced by the parallel classes of rhe KTS( u). We shal describe the subsystem as a sub-KTS( ur) . The obvious nccessary conditions for the existenceofa KTS(u) containing a sub-KTS(u) is u ) 3 w, v : 1t) : 3 modulo 6. This problem has been studied in several recent papers, antl the following results have been proved.