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

The upper chromatic number View the MathML sourceχ¯(H) of a hypergraph H=(X,E)H=(X,E) is the maximum number k for which there exists a partition of X into non-empty subsets X=X1∪X2∪⋯∪XkX=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⩾3n⩾3 there exists a 3-uniform hypergraph with n vertices, upper chromatic number 2 and ⌈n(n-2)/3⌉⌈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.