Gap between graph-removal-lemma bounds

Narrow the gap between the best known upper and lower bounds for the graph-removal-lemma parameter $\delta$ as a function of $\varepsilon$, particularly by improving the tower-type upper bound or the super-polynomial lower bound for non-bipartite forbidden graphs.

Background

The graph removal lemma states that every graph that is $\varepsilon$-far from being $F$-free contains at least $\delta|G|^{|F|}$ copies of $F$. The best known upper bounds on $1/\delta$ are of tower type, while for non-bipartite $F$ the known lower bounds are super-polynomial in $1/\varepsilon$.

The paper emphasizes that these estimates are extremely far apart. Since approximate homomorphism bounds are quantitatively linked to removal-lemma bounds, resolving this problem would also improve the quantitative understanding of approximate homomorphism theorems.