  Let $H$ be a hypergraph. For a $k$-edge coloring $c : E(H) \to \{1,...,k\}$
let $f(H,c)$ be the number of components in the subhypergraph induced by the
color class with the least number of components. Let $f_k(H)$ be the maximum
possible value of $f(H,c)$ ranging over all $k$-edge colorings of $H$. If $H$
is the complete graph $K_n$ then, trivially, $f_1(K_n)=f_2(K_n)=1$. In this
paper we prove that for $n \geq 6$, $f_3(K_n)=\lfloor n/6 \rfloor+1$ and supply
close upper and lower bounds for $f_k(K_n)$ in case $k \geq 4$. Several results
concerning the value of $f_k(K_n^r)$, where $K_n^r$ is the complete $r$-uniform
hypergraph on $n$ vertices, are also established.
