Edge coloring complete uniform hypergraphs with many components

Let H be a hypergraph. For a k-edge coloring c: E(H)→{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 fk(H) be the maximum possible value of f(H,c) ranging over all k-edge colorings of H. If H is the complete graph Kn then, trivially, f1(Kn)=f2(Kn)=1. In this paper we prove that for n⩾6,f3(Kn)=⌊n/6⌋+1 and supply close upper and lower bounds for fk(Kn) in case k⩾4. Several results concerning the value of fk(Knr), where Knr is the complete r-uniform hypergraph on n vertices, are also established.