Monochromatic components in edge-colored complete uniform hypergraphs

Abstract

Let Knr denote the complete r-uniform hypergraph on vertex set V=[n]. An f-coloring is a coloring of the edges with colors {1,2,…,f}; it defines monochromatic r-uniform hypergraphs Hi=(V,Ei) for i=1,…,f, where Ei contains the r-tuples colored by i. The connected components of hypergraphs Hi are called monochromatic components. For n>rk let f(n,r,k) denote the maximum number of colors, such that in any f-coloring of Knr, there exist k monochromatic components covering V. Moreover let f(r,k)=minn>rkf(n,r,k). A reformulation (see Gyárfás (1977) [5]) of an important special case of Ryser’s conjecture states that f(2,k)=k+1 for all k. This conjecture is proved to be true only for k≤4, so the value of f(2,5) is not known. On the contrary, in this paper we show that for r>2 we can determine f(r,k) exactly, and its value is rk.