Large monochromatic components in colorings of complete hypergraphs

Gy\'arf\'as 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.