Suppose that $L\subseteq {\cal P}(\omega)$ has the following properties:


1. $\omega \notin L$, and for $e\in L$ we have $|e|\geq 2$;
2. if $e\_1\neq e\_2 \in L$ then $|e\_1\cap e\_2|\leq 1$;
3. 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 $p(n)\cap p(n+1) \neq \emptyset$?



Existence of a path in a set of subsets of $\omega$
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 $\{n,p\}\subseteq d.$


