Monochromatic Hamiltonian Berge-cycles in colored hypergraphs

It has been conjectured that for any fixed r and sufficiently large n, there is a monochromatic Hamiltonian Berge-cycle in every (r−1)-coloring of the edges of Knr, the complete r-uniform hypergraph on n vertices. In this paper, we show that the statement of this conjecture is true with r−2 colors (instead of r−1 colors) by showing that there is a monochromatic Hamiltonian t-tight Berge-cycle in every ⌊r−2t−1⌋-edge-coloring of Knr for any fixed r>t≥2 and sufficiently large n. Also, we give a proof for this conjecture when r=4 (the first open case). These results improve the previously known results in Dorbec et al. (2008) and Gyárfás et al. (2008, 2010).