  A result of Gy\'arf\'as exactly determines the size of a largest
monochromatic component in an arbitrary $r$-coloring of the complete
$k$-uniform hypergraph $K_n^k$ when $k\geq 2$ and $r-1\leq k\leq r$. We prove a
result which says that if one replaces $K_n^k$ in Gy\'arf\'as' theorem by any
``expansive'' $k$-uniform hypergraph on $n$ vertices (that is, a $k$-uniform
hypergraph $H$ on $n$ vertices in which in which $e(V_1, \dots, V_k)>0$ for all
disjoint sets $V_1, \dots, V_k\subseteq V(H)$ with $|V_i|>\alpha$ for all $i\in
[k]$), then one gets a largest monochromatic component of essentially the same
size (within a small error term depending on $r$ and $\alpha$). As corollaries
we recover a number of known results about large monochromatic components in
random hypergraphs and random Steiner triple systems, often with drastically
improved bounds on the error terms.
  Gy\'arf\'as' result is equivalent to the dual problem of determining the
smallest maximum degree of an arbitrary $r$-partite $r$-uniform hypergraph with
$n$ edges in which every set of $k$ edges has a common intersection. In this
language, our result says that if one replaces the condition that every set of
$k$ edges has a common intersection with the condition that for every
collection of $k$ disjoint sets $E_1, \dots, E_k\subseteq E(H)$ with
$|E_i|>\alpha$ for all $i\in [k]$ there exists $e_i\in E_i$ for all $i\in [k]$
such that $e_1\cap \dots \cap e_k\neq \emptyset$, then the maximum degree of
$H$ is essentially the same (within a small error term depending on $r$ and
$\alpha$). We prove our results in this dual setting.
