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 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 Kn(r), the complete r-uniform hypergraph on n vertices.
- Colored complete uniform hypergraphs
- Monochromatic matchings
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics