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 boun...

The combinatorial parity 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 parity principle requires exponentialsize bounded-d...

Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounde...

It is well known that S1 2 cannot prove the injective weak pigeonhole principle for polynomial time functions unless RSA is insecure. In this note we investigate the provability of the surjective (dual) weak pigeonhole principle in S1 2 for provably weaker function classes. §

We give exponential size lower bounds for bounded-depth Frege proofs of variants of the bijective (‘onto’) version of the pigeonhole principle, even given additional axiom schemas for modular counting principles. As a consequence we show that for bounded-depth Frege systems the general injective version of the pigeonhole principle is exponentially more powerful than its bijective version. Furth...

Recent results established exponential lower bounds for the length of any Resolution proof for the weak pigeonhole principle. More formally, it was proved that any Resolution proof for the weak pigeonhole principle, with n holes and any number of pigeons, is of length Ω(2n ǫ ), (for a constant ǫ = 1/3). One corollary is that certain propositional formulations of the statement P 6= NP do not hav...