AI 中文总结
本文研究线性超图与n-立方体$C_t^n$的平衡上色数，针对任意线性超图给出该参数的通用上界，确定$t\geq4n-2$时$C_t^n$的精确值，并在2、3维情形下给出更小t值时的相关结果。
AI 中文摘要
超图的顶点着色被称为“平衡的”，如果各颜色类的大小相差不超过1。若超边的元素两两颜色不同，则称该超边为“彩虹超边”。本文针对任意线性超图的“平衡上色数”给出了一个通用上界，平衡上色数指的是：存在超图顶点的平衡k-着色且无彩虹超边时，k所能取到的最大整数。本文聚焦于立方体$C_t^n$，其定义为顶点是$[0,t-1]^n$中的格点、超边是t个共线点构成的集合的线性超图。本文确定了当$t\geq4n-2$时$C_t^n$的精确平衡上色数；对于更小的t值，本文给出了相关界，并在2维和3维情形下确定了该参数（仅少数例外）。
英文摘要
A coloring of the vertices of a hypergraph is called \emph{balanced} if the sizes of the color classes differ by at most one. We say that a hyperedge is \emph{rainbow} if its elements have pairwise distinct colors. In this paper, we provide a general upper bound on the \emph{balanced upper chromatic number} of arbitrary linear hypergraphs, that is, the largest integer $k$ such that there exists a balanced $k$-coloring of the vertices of the hypergraph without rainbow hyperedges. We focus on the cube $C_t^n$, defined as the linear hypergraph whose vertices are the lattice points in $[0,t-1]^n$, and whose hyperedges are the sets of $t$ collinear points. We determine the exact balanced upper chromatic number of $C_t^n$ for $t\geq 4n-2$. For smaller values of $t$, we present bounds and determine this parameter (with few exceptions) in dimensions $2$ and $3$.
