# $\aleph_0$-uniform non-bipartite linear hypergraph

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 $$\kappa\geq 2^{\aleph_0}$$?

Every $$\aleph_0$$-uniform linear hypergraph is bipartite. More generally:
Theorem. If $$H=(V,E)$$ is an $$\aleph_0$$-uniform hypergraph, and if there is a number $$n\lt\aleph_0$$ such that $$|\{e\in E:S\subseteq e\}|\le\aleph_0$$ for every $$n$$-element set $$S\subseteq V$$, then $$\chi(H)\le2$$.
The proof is by induction on $$|V|$$. If $$|V|\le\aleph_0$$ then $$|E|\le\aleph_0$$; plainly an $$\aleph_0$$-uniform hypergraph with at most $$\aleph_0$$ edges is $$2$$-colorable. For the induction step we assume that $$|V|=\kappa\gt\aleph_0$$ and that the theorem holds for hypergraphs with $$\lt\kappa$$ vertices.
We may assume that $$V=\kappa$$. For each ordinal $$\alpha\in\kappa$$ let $$V_\alpha$$ be the smallest set such that $$\alpha\subseteq V_\alpha$$ and $$\forall e\in E\ (|e\cap V_\alpha|\ge n\implies e\subseteq V_\alpha)$$. Then $$|V_\alpha|\le|\alpha|+\aleph_0\lt\kappa$$ for each $$\alpha\in\kappa$$, and $$V_\beta\subseteq V_\alpha$$ if $$\beta\le\alpha$$, and $$V_\alpha=\bigcup_{\beta\lt\alpha}V_\beta$$ if $$\alpha$$ is a limit ordinal. Let $$H_\alpha=(V_\alpha,E_\alpha)$$ where $$E_\alpha=\{e\in E:e\subseteq V_\alpha\}$$. Note that $$E_\alpha=\bigcup_{\beta\lt\alpha}E_\beta$$ if $$\alpha$$ is a limit ordinal, and $$E=\bigcup_{\alpha\lt\kappa}E_\alpha$$.
By transfinite recursion we define functions $$f_\alpha:V_\alpha\to\{0,1\}$$ so that $$f_\alpha$$ is a proper $$2$$-coloring of $$H_\alpha$$ and $$f_\beta\subseteq f_\alpha$$ whenever $$\beta\le\alpha$$. To extend $$f_\alpha$$ to $$f_{\alpha+1}$$ observe that, if $$e\in E_{\alpha+1}\setminus E_\alpha$$, then $$|e\cap V_\alpha|\lt n$$, so $$|e\cap(V_{\alpha+1}\setminus V_\alpha)|=\aleph_0$$. Since $$|V_{\alpha+1}\setminus V_\alpha|\lt\kappa$$, the inductive hypothesis will do the job. Finally, $$\bigcup_{\alpha\in\kappa}f_\alpha$$ is a proper $$2$$-coloring of $$H$$.