Robertson–Seymour Theorem - Finite Form of The Graph Minor Theorem

Finite Form of The Graph Minor Theorem

Friedman, Robertson & Seymour (1987) showed that the following theorem exhibits the independence phenomenon by being unprovable in various formal systems that are much stronger than Peano arithmetic, yet being provable in systems much weaker than ZFC:

Theorem: For every positive integer n, there is an integer m so large that if G1, ..., Gm is a sequence of finite undirected graphs,
where each Gi has size at most n+i, then GjGk for some j < k.

(Here, the size of a graph is the total number of its nodes and edges, and ≤ denotes the minor ordering.)

Read more about this topic:  Robertson–Seymour Theorem

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