We prove several results concerning the communication complexity of a collision-finding problem, each of which has applications to the complexity of cutting-plane proofs, which make inferences based on integer linear inequalities.

In particular, we prove an $\Omega(n^{1-1/k} \log k \ /2^k)$ lower bound on the $k$-party number-in-hand communication complexity of collision-finding. This implies a $2^{n^{1-o(1)}}$ lower bound on the size of tree-like cutting-planes refutations of the bit pigeonhole principle CNFs, which are compact and natural propositional encodings of the negation of the pigeonhole principle, improving on the best previous lower bound of $2^{\Omega(\sqrt{n})}$. Using the method of density-restoring partitions, we also extend that previous lower bound to the full range of pigeonhole parameters.

Finally, using a refinement of a bottleneck-counting framework of Haken and Cook and Sokolov for DAG-like communication protocols, we give a $2^{\Omega(n^{1/4})}$ lower bound on the size of fully general (not necessarily tree-like) cutting planes refutations of the same bit pigeonhole principle formulas, improving on the best previous lower bound of $2^{\Omega(n^{1/8})}$.