  Given a database, a common problem is to find the pairs or $k$-tuples of
items that frequently co-occur. One specific problem is to create a small space
"sketch" of the data that records which $k$-tuples appear in more than an
$\epsilon$ fraction of rows of the database.
  We improve the lower bound of Liberty, Mitzenmacher, and Thaler [LMT14],
showing that $\Omega(\frac{1}{\epsilon}d \log (\epsilon d))$ bits are necessary
even in the case of $k=2$. This matches the sampling upper bound for all
$\epsilon \geq 1/d^{.99}$, and (in the case of $k=2$) another trivial upper
bound for $\epsilon = 1/d$.
