Erdős–Pósa property for finite sets of colorful graphs

Characterize the finite sets of colorful graphs that have the Erdős–Pósa property, beginning with sets containing a disconnected member, for which the problem remains unresolved even for pairs of annotated colorful graphs.

Background

The paper completely characterizes which individual colorful graphs have the Erdős–Pósa property under the colorful minor relation. Section 6 explains that this characterization does not extend directly to finite sets of colorful graphs: the property can depend on relational interactions among the members of the set rather than on whether each member individually has the property.

The authors exhibit pairs of annotated colorful graphs—colorful graphs whose palettes are subsets of {1}—that lie on opposite sides of the Erdős–Pósa boundary and conclude that the first unresolved case is finite sets with a disconnected member. Thus, even the two-member annotated case is not characterized.‌