Exploring the concept of perfection in 3-hypergraphs

The natural extension of the concept of perfection in graphs to hypergraphs is to define a uniform mm-hypergraph, HH, as perfect  , if it satisfies that for every subhypergraph H′H′, χ(H′)=⌈ω(H′)m−1⌉, where χ(H′)χ(H′) and ω(H′)ω(H′) are the chromatic and clique number of H′H′, respectively.It is known that comparability graphs are perfect. In this paper we introduce the concept of comparability 3-hypergraphs (those that can be transitively oriented) with the aim of proving that these are not perfect according to the natural definition. More explicitly, we exhibit three different subfamilies of comparability 3-hypergraphs which show different behaviors in respect to the relationship between the chromatic number and the clique number.