四元组的显式Baranyai划分，第一部分：四元组构造
Explicit Baranyai Partitions for Quadruples, Part I: Quadrupling Constructions
摘要：
当$k$整除$n$时，$n$顶点上的完全$k$ -一致超图可被整除为不相交的完美匹配.等价地，$n$ -集合的$k$-子集的集合可以被划分成并行类，使得每个并行类是$n$ -集合的一个划分。这个结果被称为Baranyai定理，它保证了\emph{Baranyai partitions}的存在性。不幸的是，Baranyai定理的证明使用了网络流参数，使得这个结果不明确。特别地，没有已知的方法来产生在时间和空间上的Baranyai分区，其与超图中的超边的数量线性地缩放。对于某些应用程序来说，希望具有在线性时间内生成Baranyai分区的显式构造。这种有效的构造对于$k=2$和$k=3$是已知的。本文给出了$k=4$和$n=4t$的一个显式递归四倍结构，其中$t \equiv 0,3,4,6,8,9 ~(\text{mod}~12)$ .在后续论文（第II部分）中，将考虑~ $t$的其他值，即$t \equiv 1,2,5,7,10,11 ~(\text{mod}~12)$。
英文摘要：
It is well known that, whenever $k$ divides $n$, the complete $k$-uniform hypergraph on $n$ vertices can be partitioned into disjoint perfect matchings. Equivalently, the set of $k$-subsets of an $n$-set can be partitioned into parallel classes so that each parallel class is a partition of the $n$-set. This result is known as Baranyai's theorem, which guarantees the existence of \emph{Baranyai partitions}. Unfortunately, the proof of Baranyai's theorem uses network flow arguments, making this result non-explicit. In particular, there is no known method to produce Baranyai partitions in time and space that scale linearly with the number of hyperedges in the hypergraph. It is desirable for certain applications to have an explicit construction that generates Baranyai partitions in linear time. Such an efficient construction is known for $k=2$ and $k=3$. In this paper, we present an explicit recursive quadrupling construction for $k=4$ and $n=4t$, where $t \equiv 0,3,4,6,8,9 ~(\text{mod}~12)$. In a follow-up paper (Part II), the other values of~$t$, namely $t \equiv 1,2,5,7,10,11 ~(\text{mod}~12)$, will be considered.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
