  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 $3/4$.
