A sequence of cardinal characteristics constructed with hypergraph coloring

Question

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 $n\geq 2$?

Answer 1

The cardinals $\bf k_n$ ($2\le n\lt\omega$) are all equal.

Lemma. Let $\kappa$ be an infinite cardinal. Given a set $A\subseteq[\omega]^\omega$ with $|A|=\kappa$ and $\chi(\omega,A)\gt n$, we can construct a set $B\subseteq[\omega]^\omega$ with $|B|=\kappa$ and $\chi(\omega,B)\gt n^2$.

Proof. For each $a\in A$ choose a collection $B_a\subseteq[a]^\omega$ so that the hypergraph $(a,B_a)$ is isomorphic to $(\omega,A)$, and let $B=\bigcup_{a\in A}B_a$.

Let $(\omega,B)$ be colored with $n^2$ colors; let $[n]\times[n]$ be the set of colors, and let $x\mapsto(f(x),g(x))$ be the coloring. Since $\chi(\omega,A)\gt n$, there is a set $a\in A$ such that $f$ is constant on $a$. Then, since $\chi(a,B_a)\gt n$, there is a set $b\in B_a$ such that $g$ is constant on $b$.

Corollary. If there is a set $A\subseteq[\omega]^\omega$ with $|A|=\kappa$ and $\chi(\omega,A)\gt2$, then there is a set $B\subseteq[\omega]^\omega$ with $|B|=\kappa$ and $\chi(\omega,B)=\aleph_0$.