Minimum degree and the graph removal lemma

TL;DR

This paper shows that imposing a minimum degree condition on a graph allows for a linear relationship between the number of edge removals and the number of forbidden subgraphs, simplifying the bounds in the graph removal lemma.

Contribution

It establishes the threshold minimum degree for linear bounds in the graph removal lemma and explores its effects for various graphs beyond cliques.

Findings

Linear bounds are achievable above the minimum degree threshold.

Below the threshold, bounds revert to super-polynomial complexity.

The results extend to graphs other than cliques.

Abstract

The clique removal lemma says that for every $r \geq 3$ and $\varepsilon>0$, there exists some $\delta>0$ so that every $n$-vertex graph $G$ with fewer than $\delta n^r$ copies of $K_r$ can be made $K_r$-free by removing at most $\varepsilon n^2$ edges. The dependence of $\delta$ on $\varepsilon$ in this result is notoriously difficult to determine: it is known that $\delta^{-1}$ must be at least super-polynomial in $\varepsilon^{-1}$, and that it is at most of tower type in $\log \varepsilon^{-1}$. We prove that if one imposes an appropriate minimum degree condition on $G$, then one can actually take $\delta$ to be a linear function of $\varepsilon$ in the clique removal lemma. Moreover, we determine the threshold for such a minimum degree requirement, showing that above this threshold we have linear bounds, whereas below the threshold the bounds are once again super-polynomial, as in the unrestricted removal lemma. We also investigate this question for other graphs besides cliques, and prove some general results about how minimum degree conditions affect the bounds in the graph removal lemma.