  We prove that there exist perfect graphs of arbitrarily large
clique-chromatic number. These graphs can be obtained from cobipartite graphs
by repeatedly gluing along cliques. This negatively answers a question raised
by Duffus, Sands, Sauer, and Woodrow in [Two-coloring all two-element maximal
antichains, J. Combinatorial Theory, Ser. A, 57 (1991), 109-116].
