We prove that the class of communication problems with public-coin randomized constant-cost protocols, called $BPP^0$, does not contain a complete problem. In other words, there is no randomized constant-cost problem $Q \in BPP^0$, such that all other problems $P \in BPP^0$ can be computed by a constant-cost deterministic protocol with access to an oracle for $Q$. We also show that the $k$-Hamming Distance problems form an infinite hierarchy within $BPP^0$. Previously, it was known only that Equality is not complete for $BPP^0$. We introduce a new technique, using Ramsey theory, that can prove lower bounds against arbitrary oracles in $BPP^0$, and more generally, we show that $k$-Hamming Distance matrices cannot be expressed as a Boolean combination of any constant number of matrices which forbid large Greater-Than subproblems.

翻译：我们证明了一类具有公开硬币随机常数成本协议的通信问题（称为$BPP^0$）不包含完全问题。换言之，不存在随机常数成本问题$Q \in BPP^0$，使得所有其他问题$P \in BPP^0$都能通过可访问$Q$预言机的常数成本确定性协议来计算。我们还证明了$k$-汉明距离问题在$BPP^0$内构成一个无限层级结构。此前仅已知相等性不是$BPP^0$的完全问题。我们引入了一种使用拉姆齐理论的新技术，能够证明$BPP^0$中任意预言机下的下界，更一般地，我们证明了$k$-汉明距离矩阵无法表示为任何常数数量矩阵的布尔组合，这些矩阵禁止出现大的大于子问题。