# set theory – A sequence of cardinal characteristics constructed with hypergraph coloring

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 $$ngeq 2$$?