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The Induced Removal Lemma in Sparse Graphs

Part of: Extremal combinatorics

Published online by Cambridge University Press:  30 September 2019

Shachar Sapir  and
Asaf Shapira
Shachar Sapir
Affiliation:
School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel
Asaf Shapira*
Affiliation:
School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel
*
*Corresponding author. Email: asafico@tau.ac.il
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Abstract

The induced removal lemma of Alon, Fischer, Krivelevich and Szegedy states that if an n-vertex graph G is ε-far from being induced H-free then G contains δH(ε) · nh induced copies of H. Improving upon the original proof, Conlon and Fox proved that 1/δH(ε)is at most a tower of height poly(1/ε), and asked if this bound can be further improved to a tower of height log(1/ε). In this paper we obtain such a bound for graphs G of density O(ε). We actually prove a more general result, which, as a special case, also gives a new proof of Fox’s bound for the (non-induced) removal lemma.

MSC classification

Primary: 05D99: None of the above, but in this section
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 1 , January 2020 , pp. 153 - 162
Copyright
© Cambridge University Press 2019 

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Footnotes

†

Supported in part by ISF Grant 1028/16 and ERC Starting Grant 633509.

References

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