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 $|I\cap V_i| = |I|/k$ for each $i$, 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 match those of recent work of Chakraborti for $k = 2$.

翻译：设 $k$-一致超图 $H = (V, E)$ 为 $k$-部图，若 $V$ 可划分为 $k$ 个集合 $V_1, \ldots, V_k$，使得每条边 $e \in E$ 恰好包含每个 $V_i$ 中的一个顶点。当每个 $|V_i| = n$ 时，称该超图为 $n$-平衡的。$H$ 中的独立集 $I$ 称为平衡的，若对每个 $i$ 有 $|I \cap V_i| = |I|/k$；染色称为平衡的，若每个色类在 $H$ 中导出一个平衡独立集。本文给出了关于平均度 $D = |E|/n$ 的平衡独立数 $\alpha_b(H)$ 的下界，以及关于最大度 $\Delta$ 的平衡色数 $\chi_b(H)$ 的上界。对于 $k = 2$ 的情形，我们的结果与 Chakraborti 近期的工作结论一致。