On 3‐graphs with no four vertices spanning exactly two edges

Abstract

Let D 2 $D_2$ 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 $D_2$ having polynomial bounds. We also prove an Erdős–Hajnal-type result: Every induced D 2 $D_2$ -free hypergraph on n $n$ vertices contains a clique or an independent set of size n c $n^{c}$ for some absolute constant c > 0 $c > 0$ . In the case of both problems, D 2 $D_2$ is the only nontrivial k $k$ -uniform hypergraph with k ⩾ 3 $k\geqslant 3$ which admits a polynomial bound.