Multiparty Communication Complexity of Collision Finding

We prove an $\mathrm{\Omega }(n^{1-1\mathrm{/}k}\log k\mathrm{/}{2}^k)$ 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 proofs of the bit pigeonhole principle, a compact and natural propositional encoding of the pigeonhole principle, improving on the best previous lower bound of $2^{\mathrm{\Omega }(\sqrt{n})}$.