We study the problem of extracting randomness from somewhere-random sources, and related combinatorial phenomena: partition analogues of Shearer's lemma on projections. A somewhere-random source is a tuple $(X_1, \ldots, X_t)$ of (possibly correlated) $\{0,1\}^n$-valued random variables $X_i$ where for some unknown $i \in [t]$, $X_i$ is guaranteed to be uniformly distributed. An $extracting$ $merger$ is a seeded device that takes a somewhere-random source as input and outputs nearly uniform random bits. We study the seed-length needed for extracting mergers with constant $t$ and constant error. We show: $\cdot$ Just like in the case of standard extractors, seedless extracting mergers with even just one output bit do not exist. $\cdot$ Unlike the case of standard extractors, it $is$ possible to have extracting mergers that output a constant number of bits using only constant seed. Furthermore, a random choice of merger does not work for this purpose! $\cdot$ Nevertheless, just like in the case of standard extractors, an extracting merger which gets most of the entropy out (namely, having $\Omega$ $(n)$ output bits) must have $\Omega$ $(\log n)$ seed. This is the main technical result of our work, and is proved by a second-moment strengthening of the graph-theoretic approach of Radhakrishnan and Ta-Shma to extractors. In contrast, seed-length/output-length tradeoffs for condensing mergers (where the output is only required to have high min-entropy), can be fully explained by using standard condensers. Inspired by such considerations, we also formulate a new and basic class of problems in combinatorics: partition analogues of Shearer's lemma. We show basic results in this direction; in particular, we prove that in any partition of the $3$-dimensional cube $[0,1]^3$ into two parts, one of the parts has an axis parallel $2$-dimensional projection of area at least $3/4$.

翻译：我们研究从某处随机源中提取随机性的问题，以及相关的组合现象：Shearer引理在投影上的分区类比。某处随机源是一个由（可能相关的）$\{0,1\}^n$值随机变量$X_i$组成的元组$(X_1, \ldots, X_t)$，其中对于某个未知的$i \in [t]$，$X_i$保证是均匀分布的。一个提取合并器是一种有种子装置，它以某处随机源为输入，输出接近均匀的随机比特。我们研究在常数$t$和常数误差下提取合并器所需的种子长度。我们证明：· 与标准提取器的情况一样，即使只有一个输出比特的无种子提取合并器也不存在。· 与标准提取器的情况不同，仅使用常数种子就可以实现输出常数比特数的提取合并器。此外，随机选择的合并器无法实现这一目的！· 尽管如此，与标准提取器的情况一样，一个提取大部分熵（即输出$\Omega(n)$比特）的提取合并器必须具有$\Omega(\log n)$的种子。这是我们工作的主要技术结果，通过对Radhakrishnan和Ta-Shma提取器图论方法的二阶矩加强来证明。相比之下，压缩合并器（仅要求输出具有高最小熵）的种子长度与输出长度权衡可以通过使用标准压缩器完全解释。受这些考虑的启发，我们还提出了组合学中的一个新的基本问题类别：Shearer引理在投影上的分区类比。我们展示了这方面的基本结果；特别地，我们证明在将$3$维立方体$[0,1]^3$任意划分为两部分时，其中一部分在某个轴平行的$2$维投影中的面积至少为$3/4$。