Monochromatic components in edge-colored complete uniform hypergraphs

Abstract

Abstract Let K n r 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 H i = ( V , E i ) for i = 1 , … , f , where E i contains the r-tuples colored by i. The connected components of hypergraphs H i are called monochromatic components. For n > r k let f ( n , r , k ) denote the maximum number of colors, such that in any f-coloring of K n r , there exist k monochromatic components covering V. Moreover let f ( r , k ) = min n > r k f ( n , r , k ) . A reformulation (see [A. Gyarfas Partition coverings and blocking sets in hypergraphs, Commun. Comput. Autom. Inst. Hungar. Acad. Sci. 71 (1977)]) of an important special case of Ryserʼs 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.