Polynomial removal lemmas for ordered graphs

Lior Gishboliner; István Tomon

doi:10.5070/C62359151

Polynomial removal lemmas for ordered graphs

Lior Gishboliner, István Tomon

Research output: Contribution to journal › Article › peer-review

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 ε > 0, then there exists δ_F (ε) > 0 such that every n-vertex ordered graph G containing at most δ_F (ε)n^v(F) induced copies of F can be made induced F-free by adding/deleting at most εn² edges. We prove that δ_F (ε) can be chosen to be a polynomial function of ε 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 noninduced case.

Original language	English
Article number	#3
Journal	Combinatorial Theory
Volume	2
Issue number	3
DOIs	https://doi.org/10.5070/C62359151
State	Published - 2022
Externally published	Yes

Keywords

Ordered graph
removal lemma

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10.5070/C62359151

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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 ε > 0, then there exists δF (ε) > 0 such that every n-vertex ordered graph G containing at most δF (ε)nv(F) induced copies of F can be made induced F-free by adding/deleting at most εn2 edges. We prove that δF (ε) can be chosen to be a polynomial function of ε 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 noninduced case.",

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N2 - 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 ε > 0, then there exists δF (ε) > 0 such that every n-vertex ordered graph G containing at most δF (ε)nv(F) induced copies of F can be made induced F-free by adding/deleting at most εn2 edges. We prove that δF (ε) can be chosen to be a polynomial function of ε 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 noninduced case.

AB - 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 ε > 0, then there exists δF (ε) > 0 such that every n-vertex ordered graph G containing at most δF (ε)nv(F) induced copies of F can be made induced F-free by adding/deleting at most εn2 edges. We prove that δF (ε) can be chosen to be a polynomial function of ε 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 noninduced case.

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Polynomial removal lemmas for ordered graphs

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