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

Abstract

The upper chromatic number over(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 = X1 ∪ X2 ∪ ⋯ ∪ Xk such that for each edge at least two vertices lie in one of the partite sets. We prove that for every n {greater than or slanted equal to} 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. © 2006 Elsevier B.V. All rights reserved.