We show that every resolution proof of the functional version FPHPm n of the pigeonhole principle (in which one pigeon may not split between several holes) must have size exp ( Ω ( n (logm)2 )) . This implies an exp ( Ω(n1/3) ) bound when the number of pigeons m is arbitrary.

We prove lower bounds of the form exp (n " d) ; " d > 0; on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d; for any constant d: This is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known.

We extend results of Bonet, Buss and Pitassi on Bondy’s Theorem and of Nozaki, Arai and Arai on Bollobás’ Theorem by proving that Frankl’s Theorem on the trace of sets has quasipolynomial size Frege proofs. For constant values of the parameter t, we prove that Frankl’s Theorem has polynomial size AC-Frege proofs from instances of the pigeonhole principle. §

The use of classical logic for some combinatorial proofs, as it is the case with Ramsey’s theorem, can be localized in the Infinite Pigeonhole (IPH) principle, stating that any infinite sequence which is finitely colored has an infinite monochromatic subsequence. Since in general there is no computable functional producing such an infinite subsequence, we consider a Π2-corollary, proving the cl...

Cook and Reckhow proved in 1979 that the propositional pigeonhole principle has polynomial size extended Frege proofs. Buss proved in 1987 that it also has polynomial size Frege proofs; these Frege proofs used a completely different proof method based on counting. This paper shows that the original Cook and Reckhow extended Frege proofs can be formulated as quasipolynomial size Frege proofs. Th...

We see that the version of the pigeonhole principle in which every hole is forced to receive a pigeon (called onto) and the version in which every pigeon is mapped into exactly one hole (called functional) have polynomial-size proofs in the tree-like monotone sequent calculus. The proofs are surprisingly simple reductions to the non-monotone case.

We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(PV )), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP , APP , MA, AM ) in PV1 + dWPHP(PV ).

We give a proof of the classical Marriage Lemma 4] using completeness of hyperresolution. This argument is purely syntactical, and extends directly to the innnite case. As an application we give a purely syntactical version of a proof that resolution is exponential on the pigeonhole principle.

During the last decade, an active line of research in proof complexity has been to study space complexity and timespace trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving. For the polynomial calculus proof system, the only previously known space lower bound is for CNF formulas of unbounded width in [Alekhnovic...