Gyárfás famously showed that in every r-coloring of the edges of the complete graph $K_n$, there is a monochromatic connected component with at least $\frac{n}{r-1}$ vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for k-uniform hypergraphs. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of $1+o(1)$ as the number of colors grows. We further generalize these questions to ask about counts of vertex s-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.

Gyárfás 著名地证明了在完整图的边缘的每个r着色中$K_n$，有一个单色连通分量至少有$\frac{n}{r-1}$顶点。Conlon、Tyomkyn 和第二作者最近的一项研究解决了有关具有许多边的单色连通分量的类似问题。在本文中，我们研究了k均匀超图的这些问题的概括。通过将连通性定义广泛扩展至更高的均匀性，我们提供了最大单色分量尺寸的上限和下限，该上限和下限严格限制为$1+哦（1）$随着颜色数量的增加。我们进一步概括这些问题来询问大型单色组件的边缘中包含的顶点s集的数量。在两种颜色的特定情况下，我们得出更精确的结果。