  A result of Gy\'arf\'as says that for every $3$-coloring of the edges of the
complete graph $K_n$, there is a monochromatic component of order at least
$\frac{n}{2}$, and this is best possible when $4$ divides $n$. Furthermore, for
all $k\geq 3$ and every $(k+1)$-coloring of the edges of the complete
$k$-uniform hypergraph $K_n^{k}$, there is a monochromatic component of order
at least $\frac{kn}{k+1}$ and this is best possible for all $n$.
  Recently, Guggiari and Scott and independently Rahimi proved a strengthening
of the graph case in the result above which says that the same conclusion holds
if $K_n$ is replaced by any graph on $n$ vertices with minimum degree at least
$\frac{5n}{6}-1$; furthermore, this bound on the minimum degree is best
possible.
  We prove a strengthening of the $k\geq 3$ case in the result above which says
that the same conclusion holds if $K_n^k$ is replaced by any $k$-uniform
hypergraph on $n$ vertices with minimum $(k-1)$-degree at least
$\frac{kn}{k+1}-(k-1)$; furthermore, this bound on the $(k-1)$-degree is best
possible.
