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 K n ( r ) , 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 K n ( r ) for large n; and a Berge-cycle of order ( 1 − o ( 1 ) ) n in ( r − ⌊ log 2 r ⌋ ) -colorings of K n ( r ) . 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 K n ( r ) .