article published yes Jacob Fox author Yuval Wigderson author 2d0023a0-1567-11f0-833d-d5c1e476d4b5 The clique removal lemma says that for every ≥r 3 andε > 0, there exists some δ > 0 so that every n‐vertex graph G with fewer than δnr copies of K r can be made K r ‐free by removing at most εn2 edges. The dependence of δ on ε in this result is notoriously difficult to determine: it is known that δ−1 must be at least super‐polynomial in ε−1, and that it is at most of tower type in εlog −1. We prove that if one imposes an appropriate minimum degree condition on G, then one can actually take δ to be a linear function of ε 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. Wiley2023 eng chromatic thresholdgraph removal lemmahomomorphism thresholdminimum degree conditions 0364-9024 1097-0118 2105.0919410.1002/jgt.22891 1024648-665 yes J. Fox, Y. Wigderson, Journal of Graph Theory 102 (2023) 648–665. Fox, J., &#38; Wigderson, Y. (2023). Minimum degree and the graph removal lemma. <i>Journal of Graph Theory</i>. Wiley. <a href="https://doi.org/10.1002/jgt.22891">https://doi.org/10.1002/jgt.22891</a> Fox J, Wigderson Y. 2023. Minimum degree and the graph removal lemma. Journal of Graph Theory. 102(4), 648–665. Fox J, Wigderson Y. Minimum degree and the graph removal lemma. <i>Journal of Graph Theory</i>. 2023;102(4):648-665. doi:<a href="https://doi.org/10.1002/jgt.22891">10.1002/jgt.22891</a> J. Fox and Y. Wigderson, “Minimum degree and the graph removal lemma,” <i>Journal of Graph Theory</i>, vol. 102, no. 4. Wiley, pp. 648–665, 2023. Fox, Jacob, and Yuval Wigderson. “Minimum Degree and the Graph Removal Lemma.” <i>Journal of Graph Theory</i>. Wiley, 2023. <a href="https://doi.org/10.1002/jgt.22891">https://doi.org/10.1002/jgt.22891</a>. Fox, Jacob, and Yuval Wigderson. “Minimum Degree and the Graph Removal Lemma.” <i>Journal of Graph Theory</i>, vol. 102, no. 4, Wiley, 2023, pp. 648–65, doi:<a href="https://doi.org/10.1002/jgt.22891">10.1002/jgt.22891</a>. 221642026-06-29T10:53:26Z2026-07-14T08:24:20Z