If $H=(V,E)$ is a hypergraph and $\kappa$ is a cardinal,we say a map $c:V\to\kappa$ is a *coloring* if the restriction $c\restriction_e$ of $c$ to $e$ is non-constant whenever $e\in E$ and $|e|>1$. The smallest cardinal such that there is a coloring from $V$ to that cardinal is denoted by $\chi(H)$.

By $[\omega]^\omega$ we denote the set of infinite subsets of $\omega$.

Given $n\in\omega,n>1$ is there $E\subseteq[\omega]^\omega$ such that $\chi(\omega,E)=n$?