Cover k-Uniform Hypergraphs by Monochromatic Loose Paths

A conjecture of Gyárfás and Sárközy says that in every 2-coloring of the edges of the complete k-uniform hypergraph Kk n, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most k − 2 vertices. Recently, the authors affirmed the conjecture. In the note we show that for every 2-coloring of Kk n, one can find two monochromatic paths of distinct colors to cover all vertices of Kk n such that they share at most k − 2 vertices. Omidi and Shahsiah conjectured that R(Pk t ,Pk t ) = t(k − 1) + b t+1 2 c holds for k > 3 and they affirmed the conjecture for k = 3 or k > 8. We show that if the conjecture is true, then k−2 is best possible for our result.