On Small-depth Frege Proofs for PHP

We study Frege proofs for the one-to-one graph Pigeon Hole Principle defined on the $n\times n$ grid where $n$ is odd. We are interested in the case where each formula in the proof is a depth $d$ formula in the basis given by $\wedge$, $\vee$, and $\mathrm{\neg }$. We prove that in this situation the proof needs to be of size exponential in $n^{\mathrm{\Omega }(1\mathrm{/}d)}$. If we restrict the size of each line in the proof to be of size $M$ then the number of lines needed is exponential in $n\mathrm{/}(\log M{)}^{O(d)}$. The main technical component of the proofs is to design a new family of random restrictions and to prove the appropriate switching lemmas.