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.
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.