The existence of unavoidable sets of geographically good configurations

Abstract

A set of configurations is unavoidable if every planar map contains at least one element of the set. A configuration $\mathcal{L}$ is called geographically good if whenever a member country $M$ of $\mathcal{L}$ has any three neighbors $N_{1}$, $N_{2}$, $N_{3}$ which are not members of $\mathcal{L}$ then $N_{1}$, $N_{2}$, $N_{3}$ are consecutive (in some order) about $M$. The main result is a constructive proof that there exist finite unavoidable sets of geographically good configurations. This result is the first step in an investigation of an approach towards the Four Color Conjecture.