# Proof of Prime Ideal Theorem from De Brujin Erdös Theorem

Given the theory $$\operatorname{ZF} + \text{For every graph G, if every finite subgraph is 3-colorable, then G is 3-colorable}$$ How can I prove the Boolean Prime Ideal Theorem?

(This is problem 2.24 in Jech's book "The Axiom of Choice")

• You're right. Sorry 'bout that. Did you follow the references in the historical notes? – Asaf Karagila Dec 3 '16 at 18:17