Abstract: The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph H and ε>0, if an n-vertex graph G contains εn^2 edge-disjoint copies of H then G contains δn^{v(H)} copies of H for some δ=δ(ε,H)>0. The current proofs of the removal lemma give only very weak bounds on δ(ε,H), and it is also known that δ(ε,H) is not polynomial in ε unless H is bipartite. Recently, Fox and Wigderson initiated the study of minimum degree conditions guaranteeing that δ(ε,H) depends polynomially or linearly on ε. In this talk, we answer several questions of Fox and Wigderson on this topic. In particular, we provide the polynomial removal lemma threshold for odd cycles and give the full characterization of the linear removal lemma threshold for 3-chromatic graphs.
