论文标题
不可还原的子立方体分区
Irreducible subcube partitions
论文作者
论文摘要
a \ emph {subcube分区}是布尔立方体$ \ {0,1 \}^n $的分区。如果唯一的联合子立方体是单人群和整个分区的唯一子分区，则子立方体分区是不可约的。如果子立方体分区“提及”所有坐标，则该分区很紧。我们研究紧密不可还原子立方体分区的极端特性：最小的尺寸，最小的重量，最大点数，最大尺寸和最大最小尺寸。我们还考虑存在均匀的紧密不可还原子立方体分区的存在，其中所有子立面都具有相同的维度。我们还研究了$ \ {0，\ dots，q-1 \}^n $的子立方体分区，以及$ \ mathbb {f} _2^n $的分区中的分区，在两种情况下都集中在最小尺寸上。我们的构造和计算机实验导致对上述特性的最大值的几个猜想。
A \emph{subcube partition} is a partition of the Boolean cube $\{0,1\}^n$ into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it "mentions" all coordinates. We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible subcube partitions, in which all subcubes have the same dimensions. We additionally study subcube partitions of $\{0,\dots,q-1\}^n$, and partitions of $\mathbb{F}_2^n$ into affine subspaces, in both cases focusing on the minimal size. Our constructions and computer experiments lead to several conjectures on the extremal values of the aforementioned properties.
