Minimum degree and the graph removal lemma

The clique removal lemma says that for every (Formula presented.) and (Formula presented.), there exists some (Formula presented.) so that every (Formula presented.) -vertex graph (Formula presented.) with fewer than (Formula presented.) copies of (Formula presented.) can be made (Formula presented.) -free by removing at most (Formula presented.) edges. The dependence of (Formula presented.) on (Formula presented.) in this result is notoriously difficult to determine: it is known that (Formula presented.) must be at least super-polynomial in (Formula presented.), and that it is at most of tower type in (Formula presented.). We prove that if one imposes an appropriate minimum degree condition on (Formula presented.), then one can actually take (Formula presented.) to be a linear function of (Formula presented.) 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.