Suppose $\langle s\_\alpha : \alpha \in \omega\_2 \cap \mathrm{cof}(\omega\_1) \rangle$ is a sequence such that each $s\_\alpha$ is an increasing cofinal map from $\omega\_1$ to $\alpha$. Is it possible that for all $\alpha < \beta$, $\mathrm{ran}(s\_\alpha) \cap \mathrm{ran}(s\_\beta)$ is finite? Note that a pressing-down plus pigeonhole argument shows that CH must fail.



almost disjoint ladder system on $\omega_2$
Following the suggestion, here is the solution, from Baumgartner: Almost-disjoint sets, the dense set problem and the partition calculus, Annals of Math. Logic, 10(1976), p. 424, part 6.


Let $\{A\_\xi:\xi<\omega\_2\}$ be sets of size $\aleph\_1$ with pairwise countable intersection. 
Set $p\in P$ iff $p$ is a function, $Dom(p)\in [\omega\_2]^{<\omega}$, $p(\xi)\in [A\_\xi]^{<\omega}$ ($\xi\in Dom(p)$). $p'\leq p$ if $Dom(p')\supseteq Dom(p)$, $p'\supseteq p(\xi)$ and $p(\xi)\cap p(\eta)=p'(\xi)\cap p'(\eta)$ ($\xi\neq\eta\in Dom(p)$). 
If $G$ is generic, we set $B\_\xi=\bigcup\{p(\xi):\xi\in Dom(p),p\in G\}$. Density arguments give $|B\_\xi|=\aleph\_1$, $|B\_\xi\cap B\_\eta|<\omega$, essentially the only nontrivial point is to show that $(P,\leq)$ is ccc. 


For that, assume that $p\_\alpha\in P$ ($\alpha<\omega\_1$). By the $\Delta$-system lemma, we can assume that $Dom(p\_\alpha)=s\cup s\_\alpha$ with $\{s,s\_\alpha:\alpha<\omega\}$ disjoint. The common extension of $p\_\alpha$, $p\_\beta$ will clearly be the coordinatewise union, the only trouble can be if $p\_\alpha(\xi)\cap p\_\alpha(\eta)\neq p\_\beta(\xi)\cap p\_\beta(\eta)$ for some pair $\{\xi,\eta\}\in[s]^2$. However, as $|B\_\xi\cap B\_\eta|\leq\omega$ for $\xi\neq\eta$, there are just countably many possibilities for $p\_\alpha(\xi)\cap p\_\alpha(\eta)$ for each pair $(\xi,\eta)$, so we can find $\alpha$ and $\beta$ that $p\_\alpha(\xi)\cap p\_\alpha(\eta)=p\_\beta(\xi)\cap p\_\beta(\eta)$ for all pairs $\{\xi,\eta\}$ in $s$, and then $p\_\alpha$, $p\_\beta$ will be compatible. 


The nice thing in the argument is that it is iterable, i.e., $|A\_\alpha|=\aleph\_2$, $|A\_\alpha\cap A\_\beta|\leq\aleph\_1$ for $\alpha<\beta<\omega\_3$, then one forcing shrinks each $A\_\alpha$ to $A'\_\alpha$ so that $|A'\_\alpha|=\aleph\_2$, $|A'\_\alpha\cap A'\_\beta|\leq\aleph\_0$, then the next forcing thins out $A'\_\alpha$ to $A''\_\alpha$ such that $A''\_\alpha\cap A''\_\beta$ is finite, etc. 


