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