Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$.

A hypergraph $H=(V,E)$ consists of a set $V$ and a collection of subsets $E \subseteq {\cal P}(V)$. A *coloring* is a map $c: V\to \kappa$, where $\kappa \neq \emptyset$ is a cardinal, such that for every $e\in E$ with $|e|\geq 2$ the restriction $c|_e$ is non-constant. We denote the minimal cardinal $\kappa$ such that there is a coloring $c: V\to \kappa$ by $\chi(H)$ and call it the *chromatic number* of $H$

If $A\subseteq [\omega]^\omega$ is finite or countable, then the chromatic number of $(\omega, A)$ equals $2$. This motivates the following cardinals: for any integer $n\geq 2$ let ${\bf k}_n$ be the minimum cardinality of a set $A\subseteq [\omega]^\omega$ such that $\chi(\omega, A) > n$.

Is it consistent in ${\sf ZFC}$ that $${\bf k}_n < {\bf k}_{n+1} < 2^{\aleph_0}$$ for all integers $n\geq 2$?