How to prove a certain connection between the list-chromatic number of a bipartite graph and a cardinal? Let $G=(L,R,E)$ be a complete bipartite graph, such that $|L|=\aleph_0$ and $|R|=\kappa$. I'd like to show that if $\kappa<2^{\aleph_0}$ then the list-chromatic number of $G$ can't be more than $\aleph_0.$
Intuitively I find this correct, but I couldn't find a way to prove this. Is it really true that $\chi_l(G) = \aleph_0$? 
 A: From among the $2^{\aleph_0}$ different ways to color the vertices of $L$ (from the given lists), you can choose a family of $2^{\aleph_0}$ almost disjoint colorings. (Two colorings are almost disjoint if they have only finitely many colors in common.)

For each vertex $v\in V(G)=L\cup R$, let $C(v)$ be the list of colors associated with $v$; we assume that $|C(v)|=\aleph_0$. Let $L=\{v_n:n\lt\omega\}$ and let $C_n=C(v_n)$. Choose a color $c\in C_0$; then choose two distinct colors $c_0,c_1\in C_1\setminus \{c_0\}$; then choose four distinct colors $c_{00},c_{01},c_{10},c_{11}\in C_2\setminus\{c,c_0,c_1\}$; and so on. In this way we build a full binary tree of height $\omega$: the nodes of the tree are distinct colors, the nodes on level $n$ are colors in $C_n$, each branch is a list-coloring of $L$, there are $2^{\aleph_0}$ branches, and two different branches have only finitely many colors in common.

At least one of those colorings (in fact all but $\kappa$ of them) can be extended to a proper vertex coloring
of $G$.

For each vertex $w\in R$, its list $C(w)$, being infinite, is contained in at most one of those $2^{\aleph_0}$ almost disjoint colorings of $L$. Since $|R|\lt2^{\aleph_0}$, we can choose a coloring of $L$ which does not contain any of the lists $C(w),w\in R$. Clearly such a coloring can be extended.

That's how you show that $\chi_\ell(G)\le\aleph_0$. To see that $\chi_\ell(G)\ge\aleph_0$ observe that $\chi_\ell(G)\ge\chi_{\ell}(K_{\aleph_0,\aleph_0})\ge\chi_\ell(K_{n,n^n})=n+1$ for each $n\in\mathbb N$.

It's clear that $\chi_\ell(K_{n,n^n})\le n+1$. To show that $\chi_\ell(K_{n,n^n})\gt n$, assign pairwise disjoint lists of size $n$, call them $C_1,C_2,\dots,C_n$, to the vertices of $L$; then assign the $n^n$ different transversals of the family $\{C_1,C_2,\dots,C_n\}$ to the $n^n$ vertices of $R$.

