If $H=(V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) and $\kappa \neq \emptyset$ is a cardinal, then a map $c:V\to\kappa$ is called a *coloring* if the restriction $c\restriction\_e$ is non-constant for all $e\in E$ with $|e|\geq 2$. The smallest cardinal $\kappa$ for which a coloring $c:V\to \kappa$ exists, is said to be the *chromatic number* $\chi(H)$ of $H$.


We say $J\subseteq V$ is *independent* if $|J\cap e| \leq 1$ for all $e\in E$. We say that a collection ${\cal J}$ of independent sets is an *independent covering* of $H$ if $\bigcup {\cal J} = V$. The smallest cardinality of any independent covering of $H$ is said to be the *independent covering number* $j(H)$ of $H$.


It is not difficult to see that if $H$ is a graph, then $j(H) = \chi(H)$.


We say $H$ is $k$-regular for an integer $k\geq 2$ if $|e|=k$ for all $e\in E$.


**Question.** Given an integer $k\geq 3$ is there a positive integer $C\_k$ such that
$$j(H) \leq C\_k\cdot \chi(H)$$ for all finite $k$-regular hypergraphs $H$?



Discrepancy of chromatic number and independent covering number for $k$-regular hypergraphs
No, this is false already for $k=3$. Let $A$ and $B$ be disjoint sets of size $n$, and let $H$ be the hypergraph with vertex set $A \cup B$, whose hyperedges are all $3$-subsets $e$ of $A \cup B$ such that $|e \cap A| \in \{1, 2\}$. Note that $\chi(H)=2$, since we can colour all vertices in $A$ red and all vertices in $B$ blue. On the other hand, since every subset of size $2$ is contained in some hyperedge, the largest independent set has size $1$. Thus, $j(H)=2n$.


