 In their paper, they credited Paul Erdős with assistance in its proof.They applied these results to the comparability graphs of partially ordered sets to show that each partial order contains either a countably infinite antichain or a chain of cardinality equal to the whole order, and that each partial order contains either a countably infinite chain or an antichain of cardinality equal to the whole order.The same theorem can also be stated as a result in set theory, using the arrow notation of , as formula_1.