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 $\mathrm{an}$ 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 Baranyai partitions. Unfortunately, the proof of Baranyai's theorem uses network flow arguments, making this result nonexplicit. 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 3. In this paper, we present an explicit recursive quadrupling construction for $k=4$ and $n=4t$, where $t\equiv 0,3,4,6,8,9\left(\text{mod}12\right)$. In a follow‐up paper (Part II), the other values of $t$, namely, $t\equiv 1,2,5,7,10,11\left(\text{mod}12\right)$, will be considered.

中文翻译：

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显式的Baranyai分区为四倍，第一部分：四倍构造

众所周知，每当 $ķ$ 分界 $ñ$，完整 $ķ$-一致的超图 $ñ$顶点可以划分为不相交的完美匹配。等效地，$ķ$的子集 $\mathrm{一个}$ 可以将n- set划分为并行类，以便每个并行类都是$ñ$-放。这个结果被称为Baranyai定理，它保证了Baranyai分区的存在。不幸的是，Baranyai定理的证明使用网络流参数，从而使该结果不明确。特别是，尚无已知的方法可以在时间和空间上生成与超图中超边的数量成线性比例的Baranyai分区。对于某些应用程序，希望具有一种在线性时间内生成Baranyai分区的显式结构。这种高效的结构以$ķ=\mathrm{2个}$ 以及3.在本文中，我们为 $ķ=4$ 和 $ñ=4Ť$， 在哪里 $Ť\equiv 0，3，4，6，8，9（\text{国防部}12）$。在后续文件（第二部分）中，$Ť$，即 $Ť\equiv \mathrm{1个}，\mathrm{2个}，5，7，10，11（\text{国防部}12）$， 将被考虑。