Polynomial removal lemmas for ordered graphs

TL;DR

This paper characterizes when polynomial bounds exist in ordered graph removal lemmas, showing they occur only for very simple graphs, and extends discussion to non-induced cases.

Contribution

It establishes necessary and sufficient conditions for polynomial bounds in ordered graph removal lemmas, a significant refinement of previous results.

Findings

Polynomial bounds hold only for graphs with two vertices or a specific three-vertex structure.

The result delineates the boundary between polynomial and non-polynomial removal lemmas.

Discussion of analogous problems in the non-induced setting.

Abstract

A recent result of Alon, Ben-Eliezer and Fischer establishes an induced removal lemma for ordered graphs. That is, if $F$ is an ordered graph and $\varepsilon>0$, then there exists $\delta_{F}(\varepsilon)>0$ such that every $n$-vertex ordered graph $G$ containing at most $\delta_{F}(\varepsilon) n^{v(F)}$ induced copies of $F$ can be made induced $F$-free by adding/deleting at most $\varepsilon n^2$ edges. We prove that $\delta_{F}(\varepsilon)$ can be chosen to be a polynomial function of $\varepsilon$ if and only if $|V(F)|=2$, or $F$ is the ordered graph with vertices $x<y<z$ and edges $\{x,y\},\{x,z\}$ (up to complementation and reversing the vertex order). We also discuss similar problems in the non-induced case.