Improved bounds on the Weak Pigeonhole Principle and infinitely many primes from weaker axioms

We show that the known bounded-depth proofs of the Weak Pigeonhole Principle PHn2n in size n(O(log(n))) are not optimal in terms of size. More precisely, we give a size-depth trade-off upper bound: there are proofs of size n(O(d(log(n))2/d)) and depth O(d). This solves an open problem of Maciel et al. (Proceedings of the 32nd Annual ACM Symposium on the Theory of Computing, 2000). Our technique requires formalizing the ideas underlying Nepomnjascij's Theorem which might be of independent interest. Moreover, our result implies a proof of the unboundedness of primes in IDelta(0) with a provably weaker 'large number assumption' than previously needed. (C) 2002 Elsevier Science B.V. All rights reserved.