A $k$-uniform hypergraph $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that every edge in $E$ contains precisely one vertex from each $V_i$. We call such a graph $n$-balanced if $|V_i| = n$ for each $i$. An independent set $I$ in $H$ is balanced if $|I\cap V_i| = |I\cap V_j|$ for each $1 \leq i, j \leq k$, and a coloring is balanced if each color class induces a balanced independent set in $H$. In this paper, we provide a lower bound on the balanced independence number $\alpha_b(H)$ in terms of the average degree $D = |E|/n$, and an upper bound on the balanced chromatic number $\chi_b(H)$ in terms of the maximum degree $\Delta$. Our results recover those of recent work of Chakraborti for $k = 2$.

翻译：若一个$k$-均匀超图$H = (V, E)$的顶点集$V$可划分为$k$个子集$V_1, \ldots, V_k$，使得每条边$e \in E$在每个$V_i$中恰包含一个顶点，则称该超图为$k$-部超图。若进一步满足对所有$i$有$|V_i| = n$，则称该图为$n$-平衡超图。$H$中的独立集$I$若满足对所有$1 \leq i, j \leq k$有$|I\cap V_i| = |I\cap V_j|$，则称为平衡独立集；若一种着色方案的每个颜色类在$H$中都导出平衡独立集，则称该着色为平衡着色。本文通过平均度$D = |E|/n$给出了平衡独立数$\alpha_b(H)$的下界，并通过最大度$\Delta$给出了平衡色数$\chi_b(H)$的上界。我们的结果推广了Chakraborti近期关于$k = 2$情形的研究结论。