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 Bergecycle in every (r − 1)-coloring of the edges of K n , 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 for large n; and a Berge-cycle of order (1− o(1))n in (r − log2 r )-colorings of K n . 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 . © 2007 Elsevier Inc. All rights reserved.