Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems

In this series of papers we advance Ramsey theory of colorings over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in particular solve two problems from [CKS21]. It is shown that for every infinite cardinal $\lambda$, a strong coloring on ${\lambda }^{+}$ by $\lambda$ colors over a partition can be stretched to one with ${\lambda }^{+}$ colors over the same partition. Also, a sufficient condition is given for when a strong coloring witnessing $P{r}_1(\dots )$ over a partition may be improved to witness $P{r}_0(\dots )$. Since the classical theory corresponds to the special case of a partition with just one cell, the two results generalize pump-up theorems due to Eisworth and Shelah, respectively.