TITLE: Existence of a path in a set of subsets of $\omega$
QUESTION [6 upvotes]: Suppose that $L\subseteq {\cal P}(\omega)$ has the following properties:

$\omega \notin L$, and for $e\in L$ we have $|e|\geq 2$;
if $e_1\neq e_2 \in L$ then $|e_1\cap e_2|\leq 1$;
if $m,n\in \omega$ there is $e\in L$ such that $\{m,n\}\subseteq e$.

It is not hard to see that $L$ is countable. Is there a bijection $p:\omega\to L$ such that for all $n\in\omega$ we have $p(n)\cap p(n+1) \neq \emptyset$?

REPLY [2 votes]: Clearly, $L$ is countably infinite. Thus it will suffice to prove the following lemma, which shows that any finite path $\langle e_1,\dots,e_k,a\rangle$ in $L$ can be extended to include any new element $b\in L.$
Lemma. Given a finite set $E=\{e_1,\dots,e_k\}\subseteq L$ and $a,b\in L\setminus E,$ we can find $c,d\in L\setminus E$ (not necessarily distinct) such that $a\cap c,\ c\cap d,$ and $d\cap b$ are nonempty.
Proof. We consider two cases.
Case 1. All elements of $E$ are finite.
Choose $m\in a,\ n\in b,$ and $p\in\omega\setminus\bigcup E.$ Find $c,d\in L$ such that $\{m,p\}\subseteq c$ and $\{n,p\}\subseteq d.$
Case 2. There is an infinite element $u\in E.$
Choose $m\in a\setminus u,\ n\in b\setminus u,$ and $p\in u\setminus\bigcup(E\setminus\{u\}).$ Find $c,d\in L$ such that $\{m,p\}\subseteq c$ and $\{n,p\}\subseteq d.$