Monochromatic matchings in the shadow graph of almost complete hypergraphs

Edge colorings of r-uniform hypergraphs naturally define a multicoloring on the 2-shadow, i.e. on the pairs that are covered by hyperedges. We show that in any (r− 1)-coloring of the edges of an r-uniform hypergraph with n vertices and at least (1−2)(n r ) edges, the 2-shadow has a monochromatic matching covering all but at most o(n) vertices. This result implies that for any fixed r and sufficiently large n, there is a monochromatic Berge-cycle of length (1− o(1))n in every (r − 1)-coloring of the edges of K (r) n , the complete r-uniform hypergraph on n vertices.