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

TL;DR

This paper proves an induced removal lemma with polynomial bounds and an Erdős-Hajnal-type result for a specific 3-uniform hypergraph, addressing a question by Alon and Shapira and highlighting unique properties of this hypergraph.

Contribution

It establishes polynomial bounds for the induced removal lemma and Erdős-Hajnal-type results specifically for the hypergraph D_2, a unique case among k-uniform hypergraphs.

Findings

Polynomial bounds for the induced removal lemma for D_2

Existence of large cliques or independent sets in D_2-free hypergraphs

D_2 is the only nontrivial k-uniform hypergraph with such properties

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 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.