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-e)( *20c nr)(1-varepsilon)left( {begin{array}{*{20}c} n r end{array}}right) edges, the 2-shadow has a monochromatic matching covering all but at most o(n) vertices. This result confirms an earlier conjecture and 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{K^{(r)}_{n}}, the complete r-uniform hypergraph on n vertices.