  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 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 b r-2 / t-1 -edge coloring of Kr n 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
[2, 3, 4].
