A note on the least number of edges of 3-uniform hypergraphs with upper chromatic number 2

Abstract

The upper chromatic number χ ¯ ( H ) of a hypergraph H = ( X , E ) is the maximum number k for which there exists a partition of X into non-empty subsets X = X 1 ∪ X 2 ∪ ⋯ ∪ X k such that for each edge at least two vertices lie in one of the partite sets. We prove that for every n ⩾ 3 there exists a 3-uniform hypergraph with n vertices, upper chromatic number 2 and ⌈ n ( n - 2 ) / 3 ⌉ edges which implies that a corresponding bound proved in [K. Diao, P. Zhao, H. Zhou, About the upper chromatic number of a co-hypergraph, Discrete Math. 220 (2000) 67–73] is best-possible.