Covering 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 K_n ^k, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most k − 2 vertices. Recently, the authors armed the conjecture. In the note we show that for every 2-coloring of K_n ^k, one can find two monochromatic paths of distinct colors to cover all vertices of K_n ^k such that they share at most k − 2 vertices. Omidi and Shahsiah conjectured that R(P_t ^k, P_t ^k) = t(k − 1) + (formula presented) holds for k ≥ 3 and they armed 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.