Let $\mathcal{C}$ be a collection of subsets of $\mathbb{N}$ such that any two distinct $U,V\in\mathcal{C}$ have finite intersection $U\cap V$. Then what can be said about the cardinality of $\mathcal{C}$?
I have a construction showing $|\mathcal{C}|=\aleph_1$ is possible, but I don’t know if $|\mathcal{C}|=|\mathbb{R}|$ is possible.
My construction uses transfinite recursion and the axiom of choice. We recursively define for each countable ordinal $\alpha$ a function $f_\alpha:\mathbb{N}\to\mathbb{N}$ such that any two distinct such functions are equal finitely often. Then the corresponding sets $U_\alpha:=\{(n,f_\alpha(n)):n\in\mathbb{N}\}\subset\mathbb{N}^2$ have pairwise finite intersection and there are $\aleph_1$ of them. Since $|\mathbb{N}^2|=|\mathbb{N}|$ this suffices.
First define for $\alpha\in\mathbb{N}$ finite that $f_\alpha(k)=\alpha$ for $k\in\mathbb{N}$. Then by the axiom of choice we can choose for each infinite countable ordinal $\alpha$ a bijection $\varphi_\alpha:\mathbb{N}\to\alpha$. For $k\in\mathbb{N}$ we define $f_\alpha(k):=\min(\mathbb{N}\setminus\{f_{\varphi_\alpha(i)}(k):i<k\})$ such that $f_\alpha(k)\neq f_{\varphi_\alpha(i)}(k)$ for all $i<k$.
