On the reflection of the countable chain condition

We study the question of when an uncountable ccc topological space $X$ contains a ccc subspace of size $\aleph_1$. We show that it does if $X$ is compact Hausdorff and more generally if $X$ is Hausdorff with $\mathrm{pct}(X) \leq \aleph_1$. For each regular cardinal $\kappa$, an example is constructed of a ccc Tychonoff space of size $\kappa$ and countable pseudocharacter but with no ccc subspace of size less than $\kappa$. We also give a ccc compact $T_1$ space of size $\kappa$ with no ccc subspace of size less than $\kappa$.