Graph minors

Minor. $H$ is a minor of $G$ if $H$ can be obtained from $G$ by deleting edges/vertices and contracting edges.

Robertson–Seymour. Every infinite set of graphs contains two where one is a minor of the other (well-quasi-ordering by minors).

Consequence. For any minor-closed property $\mathcal{P}$, there is a finite set of minimal forbidden minors.

Conclusion. Planarity is a minor-closed property with forbidden minors $K_5$ and $K_{3,3}$.