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 $c: Vto kappa$, where $kappa neq emptyset$ is a cardinal, such that for every $ein E$ with $|e|geq 2$ the restriction $c|_e$ is non-constant. We denote the minimal cardinal $kappa$ such that there is a coloring $c: Vto kappa$ by $chi(H)$ and call it the *chromatic number* of $H$

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

Is it consistent in ${sf ZFC}$ that $${frak k}_n < {frak k}_{n+1} < 2^{aleph_0}$$

for all integers $ngeq 2$?