  We consider prophet inequalities subject to feasibility constraints that are
the intersection of $q$ matroids. The best-known algorithms achieve a
$\Theta(q)$-approximation, even when restricted to instances that are the
intersection of $q$ partition matroids, and with i.i.d.~Bernoulli random
variables. The previous best-known lower bound is $\Theta(\sqrt{q})$ due to a
simple construction of [Kleinberg-Weinberg STOC 2012] (which uses
i.i.d.~Bernoulli random variables, and writes the construction as the
intersection of partition matroids).
  We establish an improved lower bound of $q^{1/2+\Omega(1/\log \log q)}$ by
writing the construction of [Kleinberg-Weinberg STOC 2012] as the intersection
of asymptotically fewer partition matroids. We accomplish this via an improved
upper bound on the product dimension of a graph with $p^p$ disjoint cliques of
size $p$, using recent techniques developed in [Alon-Alweiss European Journal
of Combinatorics 2020].
