Polynomial removal lemma for ordered matchings

TL;DR

This paper proves a polynomial removal lemma for ordered matchings and extends it to hypergraphs, showing that graphs far from being $H$-free contain many copies of $H$, confirming a special case of a conjecture.

Contribution

It establishes a polynomial bound for the removal lemma for ordered matchings and generalizes it to hypergraphs, advancing understanding of extremal combinatorics.

Findings

Graphs far from $H$-free contain polynomially many copies of $H$.

The result confirms a special case of a conjecture by Tomon and the first author.

Extension of the polynomial removal lemma to hypergraphs.

Abstract

We prove that for every ordered matching $H$ on $t$ vertices, if an ordered $n$-vertex graph $G$ is $\varepsilon$-far from being $H$-free, then $G$ contains $\text{poly}(\varepsilon) n^t$ copies of $H$. This proves a special case of a conjecture of Tomon and the first author. We also generalize this statement to uniform hypergraphs.