On $3$-graphs with no four vertices spanning exactly two edges

Abstract

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 Erdős-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 $k\ge 3$ which admits a polynomial bound.