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 $2$](https://mathoverflow.net/questions/349123/are-hypergraphs-h-v-e-with-e-v-2-colorable). 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 $n\geq 2$?



A sequence of cardinal characteristics constructed with hypergraph coloring
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 $\chi(\omega,B)=\aleph\_0$.


