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 proofs of the bit pigeonhole principle, a compact and natural propositional encoding of the pigeonhole principle, improving on the best previous lower bound of $2^{\Omega(\sqrt{n})}$.

翻译：我们证明了碰撞查找的$k$方手牌数通信复杂度具有$\Omega(n^{1-1/k} \log k \ /2^k)$的下界。这蕴含了比特鸽巢原理——鸽巢原理的一种紧凑且自然的命题编码——的树状切割平面证明规模具有$2^{n^{1-o(1)}}$的下界，改进了先前最佳的$2^{\Omega(\sqrt{n})}$下界。