  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.
