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arXiv · 2110.03577

Polynomial removal lemmas for ordered graphs

Lior Gishboliner·István Tomon
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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.

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Lior Gishboliner, István Tomon. 2021-10-07. Polynomial removal lemmas for ordered graphs. https://arxiv.org/abs/2110.03577

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