Graph removal lemma holds with 1/delta being a tower type function of log(1/epsilon)

You need to know: Graph, isomorphic graphs, subgraph.

Background: A graph G is called H-free, if it does not have a subgraph isomorphic to H. Let $T:{\mathbb R}\to{\mathbb R}$ be a function defined by the rules (i) $T(1)=2$ , (ii) $T(n)=2^{T(n-1)}$ for $n=2,3,\dots$ , and (iii), for all real x, $T(x)=T(\bar{x})$ , where $\bar{x}$ is the least positive integer such that $x \leq \bar{x}$ .

The Theorem: On 10th January 2010, Jacob Fox submitted to the Annals of Mathematics a paper in which he proved the following result. For each $\epsilon>0$ and graph H on h vertices there is a $\delta=\delta(\epsilon,H)>0$ such that every graph on n vertices with at most $\delta n^h$ copies of H can be made H-free by removing at most $\epsilon n^2$ edges. Moreover, one can choose $\delta=T(5h^4\log \epsilon^{-1})^{-1}$ .

Short context: The statement of the Theorem without “moreover” part is famous graph removal lemma proved by Erdös, Frankl, and Rödl in 1986. It has many applications in graph theory and theory of algorithms. However, from their proof one can only take $\delta \approx T(P(\epsilon^{-1}))^{-1}$ for some polynomial P. The Theorem establishes the graph removal lemma with better dependence of $\delta$ from $\epsilon$ .