The Minimum Degree Removal Lemma Thresholds

TL;DR

This paper investigates the minimum degree conditions in graphs that ensure polynomial or linear dependence of the removal lemma's parameters on epsilon, addressing open questions in extremal graph theory.

Contribution

It provides new results answering questions about degree thresholds that guarantee polynomial or linear bounds in the graph removal lemma.

Findings

Identifies degree conditions for polynomial bounds in the removal lemma.

Answers open questions posed by Fox and Wigderson.

Advances understanding of extremal conditions in graph theory.

Abstract

The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph $H$ and $\varepsilon > 0$, if an $n$-vertex graph $G$ contains $\varepsilon n^2$ edge-disjoint copies of $H$ then $G$ contains $\delta n^{v(H)}$ copies of $H$ for some $\delta = \delta(\varepsilon,H) > 0$. The current proofs of the removal lemma give only very weak bounds on $\delta(\varepsilon,H)$, and it is also known that $\delta(\varepsilon,H)$ is not polynomial in $\varepsilon$ unless $H$ is bipartite. Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that $\delta(\varepsilon,H)$ depends polynomially or linearly on $\varepsilon$. In this paper we answer several questions of Fox and Wigderson on this topic.