# Discrepancy of chromatic number and independent covering number for $k$-regular hypergraphs

If $$H=(V,E)$$ is a 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 $$H$$?

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, $$j(H)=2n$$.