**MSC:**- 05C65 Hypergraphs
- 05C15 Chromatic theory of graphs and maps
- 05B40 Packing and covering, See also {11H31, 52C15, 52C17}
- 05B07 Triple systems

vertices are the points of $P$ and whose edges are the maximal intervals in $P$.

We verify in the present paper the NP-completeness of the determination of the

(strong) chromatic number $\gamma(\mathfrak H(P))$. We give some exact or asymptotic

values of $\gamma(\mathfrak H(P))$ if $P$ is the subposet of the Boolean lattice

induced by consecutive levels $P_{l} \cup \cdots \cup P_{u}$. Moreover, we

determine $\gamma(\mathfrak H(P))$ for interval orders $P$.