Rado's conjecture (one of many equivalent formulations) states: any non-special tree has a non-special subtree of cardinality $\aleph\_1$. 


"Special" means a tree can be decomposed into countably many antichains. Hence in particular, special trees have no cofinal branches of uncountable length.


I'm asking for a reference of the consistency of RC with not CH. The usual Levy collapse of a supercompact cardinal is good for RC + CH, and the key observation is that no countably closed forcing can specialize a tree of height $\omega\_1$. With reals being added I'm not so sure. 



Consistency of Rado's conjecture with not CH
Rado's conjecture holds in Mitchell's model (of course, start with a strongly compact instead of a weakly compact) granted the following: If $T$ if a non-special tree of height $\omega\_1$, then $T$ remains non-special after Cohen forcing $Add(\omega, \kappa)$ for any regular $\kappa$. It suffices to show the case where $T$ has size $\aleph\_1$ and $\kappa=\omega\_1$ (they are in fact equivalent). We will show this.


We might assume $T$ is branchless and $\kappa$ is uncountable as if there is a branch through $T$, $T$ can't be special in any $\omega\_1$-preserving extension and if $\kappa=\omega$, then we can cook up a function in the ground as the size of the forcing is countable.


Now suppose for the sake of contradiction, $\dot{g}: T\to \omega$ is forced to be a specializing function. For each $t\in T$, let $p\_t\in Add(\omega,\kappa)$ be a condition deciding the value of $\dot{g}(t)$ to be $n\_t\in \omega$. We might shrink to a non-special subtree $T'$ of $T$ such that there exists $n\in \omega$, for all $t\in T'$, $n\_t=n$. We will develop a kind of $\Delta$-system lemma on non-special trees, aka the goal is to find non-special subtree $T''$ of $T'$ such that any $t,t'\in T''$ such that $t<t'$, $p\_t$ and $p\_{t'}$ are compatible. This will be the desired contradiction. 


First consider $C=\{t\in T': \forall s\sqsubset t \ s\neq t, dom(p\_s)\subset ht(t)\}$ ($ht$ is the height function computed in $T'$). $C$ is non-special (this can be seen using the pressing down lemma for non-speical trees by Todorcevic). 
Now consider the following pressing down function on $C$, for $t\in C$ if it is a successor, map it to its predecessor in $C$, it is a limit, then map it to $s\sqsubset t$ such that $dom(p\_t)\cap ht(t)\subset ht(s)$. Now by pressing down lemma for trees, we get non-special $T^\*\leq C$ such that the mapping is constant, say $s\in C$. Shrink further we can get a non-special $T''$ subtree of $T^\*$ and a finite partial function $r: ht(s)\to 2$ such that for all $t\in T''$, $p\_t\restriction ht(t)= r$. Fix $t<t'\in T''$, we need to show $p\_t, p\_{t'}$ are compatible. But this is clear as $p\_t\restriction ht(t)=p\_{t'}\restriction ht(t')$ and $dom(p\_t)\subset ht(t')$. 


