It is challenging to give a combinatorial condition on $K$ and a topological condition $(X,A)$ which guarantee the polyhedral product $Z_K(X,A)$ to satisfy a given topological property. In this talk, we give necessary and sufficient conditions on $K$ and $(X,A)$ for $Z_K(X,A)$ having finiteness in (generalized) Postnikov decompositions.

This is joint work with Kouyemon Iriye and Ran Levi.