The Minimum Degree Removal Lemma Thresholds

The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph $H$ and $\varepsilon > 0$, if an $n$-vertex graph $G$ contains $\varepsilon n^2$ edge-disjoint copies of $H$ then $G$ contains $\delta n^{v(H)}$ copies of $H$ for some $\delta = \delta(\varepsilon,H) > 0$. The current proofs of the removal lemma give only very weak bounds on $\delta(\varepsilon,H)$, and it is also known that $\delta(\varepsilon,H)$ is not polynomial in $\varepsilon$ unless $H$ is bipartite. Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that $\delta(\varepsilon,H)$ depends polynomially or linearly on $\varepsilon$. In this paper we answer several questions of Fox and Wigderson on this topic.