Quantitative gap in the graph removal lemma

Narrow the gap between the known tower-type upper bounds and super-polynomial lower bounds for the dependence of the graph removal lemma parameter $\delta$ on $\varepsilon$, thereby determining substantially sharper quantitative bounds.

Background

The graph removal lemma states that graphs far from being $F$-free contain many copies of $F$. The paper notes that the best known upper bounds on $1/\delta$ are of tower type, while for non-bipartite $F$ the best known lower bounds are super-polynomial in $1/\varepsilon$.

Because approximate homomorphism bounds are closely related to removal-lemma bounds, resolving this quantitative gap is also relevant to understanding the growth of $M_F(\varepsilon)$ and its asymmetric variants.