Minimum degree and the graph removal lemma

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