  The multiplayer promise set disjointness is one of the most widely used
problems from communication complexity in applications. In this problem there
are $k$ players with subsets $S^1, \ldots, S^k$, each drawn from $\{1, 2,
\ldots, n\}$, and we are promised that either the sets are (1) pairwise
disjoint, or (2) there is a unique element $j$ occurring in all the sets, which
are otherwise pairwise disjoint. The total communication of solving this
problem with constant probability in the blackboard model is $\Omega(n/k)$.
  We observe for most applications, it instead suffices to look at what we call
the ``mostly'' set disjointness problem, which changes case (2) to say there is
a unique element $j$ occurring in at least half of the sets, and the sets are
otherwise disjoint. This change gives us a much simpler proof of an
$\Omega(n/k)$ randomized total communication lower bound, avoiding Hellinger
distance and Poincare inequalities. Using this we show several new results for
data streams:
  \begin{itemize} \item for $\ell_2$-Heavy Hitters, any $O(1)$-pass streaming
algorithm in the insertion-only model for detecting if an $\eps$-$\ell_2$-heavy
hitter exists requires $\min(\frac{1}{\eps^2}\log \frac{\eps^2n}{\delta},
\frac{1}{\eps}n^{1/2})$ bits of memory, which is optimal up to a $\log n$
factor. For deterministic algorithms and constant $\eps$, this gives an
$\Omega(n^{1/2})$ lower bound, improving the prior $\Omega(\log n)$ lower
bound. We also obtain lower bounds for Zipfian distributions. \item for
$\ell_p$-Estimation, $p > 2$, we show an $O(1)$-pass $\Omega(n^{1-2/p}
\log(1/\delta))$ bit lower bound for outputting an $O(1)$-approximation with
probability $1-\delta$, in the insertion-only model. This is optimal, and the
best previous lower bound was $\Omega(n^{1-2/p} + \log(1/\delta))$.
\end{itemize}
