On calibers for $C_p(X)$

We present new results regarding calibers in the function spaces $C_p(X)$. Our main theorem is that $C_p(X)$ is strongly \v{S}anin whenever $X$ is a submetrizable space; this improves an earlier result due to Tkachuk: $C_p(X)$ is \v{S}anin whenever $X$ is a submetrizable space. Moreover, we give sufficient conditions to characterize the calibers of $C_p(X)$ when $X$ is a topological sum, and we calculate the calibers of $C_p(X)$ when $X = \prod_{\xi < \lambda}X_\xi$ is a product of non-trivial Tychonoff spaces with $i$-weight $\leq \lambda$. Furthermore, we calculate the calibers of $C_p(X)$ when $X$ is an interval of ordinals and when $X$ is the one-point $\lambda$-Lindel\"of extension of a discrete space of cardinality $\geq \lambda$. This allows to give examples of compact Hausdorff spaces $Z$ such that $iw(Z)=\kappa^{+}$ and $C_p(Z^{\kappa})$ does not have caliber $iw(Z)$; and examples of spaces $\{Z_\alpha : \alpha<cf(\kappa)\}$ such that $\kappa$ is a caliber for $C_p(Z_\alpha)$ whenever $\alpha<cf(\kappa)$ but it is not a caliber for $C_p(\bigoplus_{\alpha<cf(\kappa)} Z_\alpha)$.