An Exponential Separation between the Matching Principle and the Pigeonhole Principle

The combinatorial matching principle states that there is no perfect matching on an odd number of vertices. This principle generalizes the pigeonhole principle, which states that for a fixed bipartition of the vertices, there is no perfect matching between them. Therefore, it follows from recent lower bounds for the pigeonhole principle that the matching principle requires exponential-size bounded-depth F'rege proofs. Ajtai [AjtSO] previously showed that the matching principle does not have polynomial-size bounded-depth F'rege proofs even with the pigeonhole principle as an axiom schema. His proof utilizes nonstandard model theory and is nonconstructive. We improve Atjai's lower bound from barely superpolynomial to exponential and eliminate the nonstandard model theory. Our lower bound is also related to the inherent complexity of particular search classes (see [PapSl]). In particular, oracle separations between the complexity classes P P A and PPAD, and between P P A and P P P also follow from our tech-