Minimum degree and the graph removal lemma

Abstract

The clique removal lemma says that for every r ≥ 3 and ε > 0, there exists some δ > 0 so that every n-vertex graph G with fewer than δn r copies of K r can be made K r -free by removing at most εn 2 edges. The dependence of δ on ε in this result is notoriously difficult to determine: it is known that δ −1 must be at least superpolynomial in ε −1 , and that it is at most of tower type in log ε −1 .We prove that if one imposes an appropriate minimum degree condition on G, then one can actually take δ to be a linear function of ε 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.

Ramsey goodness of books revisited

“…The Γ constructed has high average degree but low minimum degree, and this distinction turns out to be crucial. Indeed, in [19], Fox and Wigderson proved that the K p removal lemma has linear bounds if the minimum degree of Γ is above a certain threshold, namely (1 − 2 2p−3 )m. This allows us to prove Theorem 3.1 using the technique outlined above, while obtaining much stronger quantitative control.…”

“…The first tool we need to prove Theorem 3.1 is the high-degree removal lemma with linear bounds mentioned above, from [19,Theorem 2.1]. We remark that the explicit p-dependence of the constant is not given in [19, Theorem 2.1], but it is easy to verify that the proof yields the following result.…”