Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. Let $$E = \big\{\{A,B\}: A,B \in [\omega]^\omega \text{ and } |A\cap B| \text{ is finite}\big\}.$$

We consider the graph $G=([\omega]^\omega,E)$. Maximal cliques in $G$, also known as maximal almost disjoint families, are uncountable, and it is consistent that every clique in $G$ has cardinality $< 2^{\aleph_0}$.

Is it consistent that $\chi(G) < 2^{\aleph_0}$?