On variants of conflict-free-coloring for hypergraphs

Conflict-free coloring is a kind of vertex coloring of hypergraphs requiring each hyperedge to have a color which appears only on one vertex. More generally, for a positive integer $k$ there are $k$-conflict-free colorings ($k$-CF-colorings for short) and $k$-strong-conflict-free colorings ($k$-SCF-colorings for short). %for some positive integer $k$. Let $H_n$ be the hypergraph of which the vertex-set is $V_n=\{1,2,\dots,n\}$ and the hyperedge-set $\cal{E}_n$ is the set of all (non-empty) subsets of $V_n$ consisting of consecutive elements of $V_n$. Firstly, we study the $k$-SCF-coloring of $H_n$, give the exact $k$-SCF-coloring chromatic number of $H_n$ for $k=2,3$, and present upper and lower bounds of the $k$-SCF-coloring chromatic number of $H_n$ for all $k$. Secondly, we give the exact $k$-CF-coloring chromatic number of $H_n$ for all $k$.