Monochromatic Hamiltonian Berge-cycles in colored complete uniform hypergraphs

We conjecture 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. We prove the conjecture for r=3,n⩾5 and its asymptotic version for r=4. For general r we prove weaker forms of the conjecture: there is a Hamiltonian Berge-cycle in ⌊(r−1)/2⌋-colorings of for large n; and a Berge-cycle of order (1−o(1))n in (r−⌊log2r⌋)-colorings of . The asymptotic results are obtained with the Regularity Lemma via the existence of monochromatic connected almost perfect matchings in the multicolored shadow graph induced by the coloring of .