Monochromatic loose path partitions in [formula omitted]-uniform hypergraphs

Abstract

A conjecture of Gyárfás and Sárközy says that in every 2-coloring of the edges of the complete k-uniform hypergraph Knk, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most k−2 vertices. A weaker form of this conjecture with 2k−5 uncovered vertices instead of k−2 is proved. Thus the conjecture holds for k=3. The main result of this paper states that the conjecture is true for all k≥3.