A hypergraph $H= (V,E)$ is said to be bipartite if there is $S\subseteq V$ such that for all $e\in E$ with $|e|>1$ we have $$S\cap e\neq \emptyset\neq (e\setminus S).$$ A hypergraph is said to be linear if distinct edges intersect in at most $1$ element, and we say the hypergraph is $\aleph_0$-uniform if every edge contains countably infinite elements.

If $n\geq k$ are positive integers, we denote by $n_k$ the smallest integer $n$ such that there is a $k$-uniform, linear hypergraph $(\{1,\ldots,n\}, E)$ that is not bipartite.

Question. Suppose that $\kappa\geq\aleph_0$ is a cardinal such that there is a linear, $\aleph_0$-uniform, non-bipartite hypergraph $H=(\kappa, E)$. Do we necessarily have $\kappa\geq 2^{\aleph_0}$?