Unboundedness property is ccc indestructible? I saw the following claim and I've not been able to prove it. Any suggestion is welcome.
We say $f: [\omega_2]^2\to \omega_1$ is unbounded if for any $\Gamma\in [\omega_2]^{\omega_1}$ we have $f''[\Gamma]^2$ is unbounded in $\omega_1$. The claim is the unboundedness property as above is preserved under ccc forcing. Note that such a function witnesses the failure of Chang's Conjecture. And in fact the existence of such function is equivalent to the failure of Chang's Conjecture (https://en.wikipedia.org/wiki/Chang's_conjecture).
I have some meta-mathematical justification: it shouldn't be a general fact that for any unbounded function we can find a ccc forcing which destroys the unboundedness property since we can always force $MA_{\aleph_2}$ from a model of ZFC but then if the general fact is true then $MA_{\aleph_2}$ implies CC. But we know CC has some large cardinal strength. But this is not good enough: 1) it's not direct 2) it shows there exists one unbounded function whose unboundedness property can't be destroyed by any ccc forcing, instead of any given unbounded function. 
 A: Here's an attempt at producing a counterexample.
Let $f:[\omega_2]^2 \to \omega_1$ be such that for every $X \in [\omega_2]^{\omega_1}$, $f[[X]^2]$ is unbounded in $\omega_1$. Put $g = f \upharpoonright [\omega_1]^2$.
Define a forcing $P$ whose conditions are $p = (u_p, h_p)$ where $u_p$ is a finite subset of $\omega_1$ and $h_p:[u_p]^2 \to \{0, 1\}$ and for $p, q \in P$, $p \leq q$ iff $u_q \subseteq u_p$ and $h_q = h_p \upharpoonright [u_q]^2$. Clearly, $P$ is $\omega_1$-Knaster so all cardinals are preserved. In $V^P$, define $h: [\omega_1]^2 \to \omega_1$ by 
$$
  h(\{\alpha, \beta\}) =
  \begin{cases}
    0 & \text{if $(\exists p \in G_P)(h_p(\{\alpha, \beta\}) = 0)$} \\
    g(\{\alpha, \beta\}) & \text{if $(\exists p \in G_P)(h_p(\{\alpha, \beta\}) = 1)$}
  \end{cases}
$$
Claim 1: In $V^P$, the following hold.
(a) For every finite $A \subseteq \omega_1$, there exists $\max(A) < \alpha < \omega_1$ such that for every $\beta \in A$, $h(\{\alpha, \beta\}) = 0$.
(b) For every $X \in [\omega_2]^{\omega_1}$, $f[[X]^2]$ is unbounded in $\omega_1$.
(c) For every $X \in [\omega_1]^{\omega_1}$, $h[[X]^2]$ is unbounded in $\omega_1$.
Proof of Claim 1: (a) is obvious. For (b) and (c), use the fact that $P$ is $\omega_1$-Knaster. 
In $V^P$, define $H = h \cup f \upharpoonright ([\omega_2]^2 \setminus [\omega_1]^2)$. Note that $H$ continues to satisfy: For every $X \in [\omega_2]^{\omega_1}$, $H[[X]^2]$ is unbounded in $\omega_1$.
Claim 2: There is a ccc forcing $Q$ in $V_1 = V^P$ such that $(V_1)^Q \models (\exists X \in [\omega_1]^{\omega_1}) (H \upharpoonright [X]^2 \equiv 0)$.
Proof of Claim 2: Let $Q$ consist of $v \in [\omega_1]^{< \omega}$ such that $H \upharpoonright [v]^2 \equiv 0$, ordered by reverse inclusion. In $(V_1)^Q$, define $X = \bigcup G_Q$. By Claim 1(a), $(V_1)^Q \models |X| = \omega_1$. So it suffices to show that $Q$ is ccc in $V^P$ or equivalently $P \star Q$ is ccc in $V$. Note that the set of condition $(p, v) \in P \star Q$ that satisfy: $v$ is an actual finite subset of $\omega_1$ and $v \subseteq u_p$ is dense in $P \star Q$. Suppose $\langle (p_i = (u_i, h_i), v_i) : i < \omega_1 \rangle$ is a sequence of such conditions. By thinning down, we can assume that $u_i$'s and $v_i$'s form delta-systems with roots $u_{\star}$ and $v_{\star}$ and $h_i \upharpoonright u_{\star} = h_{\star}$. But now, $(p_i, v_i)$'s are pairwise compatible. Hence $P \star Q$ is ccc.
