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