  Let $D_2$ denote the $3$-uniform hypergraph with $4$ vertices and $2$ edges.
Answering a question of Alon and Shapira, we prove an induced removal lemma for
$D_2$ having polynomial bounds. We also prove an Erd\H{o}s-Hajnal-type result:
every induced $D_2$-free hypergraph on $n$ vertices contains a clique or an
independent set of size $n^{c}$ for some absolute constant $c > 0$. In the case
of both problems, $D_2$ is the only nontrivial $k$-uniform hypergraph with
$k\geq 3$ which admits a polynomial bound.
