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 PHPn2n in size nO(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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ n^{O(d(\log (n))^{2/d} )} $$\end{document} and depth O(d). This solves an open problem of Maciel, Pitassi and Woods (2000). Our technique requires formalizing the ideas underlying Nepomnjaščij’s Theorem which might be of independent interest. Moreover, our result implies a proof of the unboundedness of primes in IΔ0 with a provably weaker ‘large number assumption’ than previously needed.