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 $H = (V, E)$ 是 $k$-分部的，若 $V$ 可划分为 $k$ 个集合 $V_1, \ldots, V_k$，使得 $E$ 中每条边恰好从每个 $V_i$ 包含一个顶点。若对每个 $i$ 有 $|V_i| = n$，则称该图为 $n$-平衡的。$H$ 中的独立集 $I$ 是平衡的，若对每个 $i$ 有 $|I\cap V_i| = |I|/k$；着色是平衡的，若每个颜色类在 $H$ 中诱导出平衡独立集。本文给出平衡独立数 $\alpha_b(H)$ 关于平均度 $D = |E|/n$ 的下界，以及平衡色数 $\chi_b(H)$ 关于最大度 $\Delta$ 的上界。我们的结果与 Chakraborti 近期关于 $k = 2$ 情形的结论相吻合。