Some earlier proofs are strengthened and refined to give the following theorem (called the blow-up lemma). Given a graph \(R\), natural number \(\Delta\), and some \(\delta>0\), there exists some \(\varepsilon>0\) that the following holds. Blow up every vertex of \(R\) to some larger set and build two graphs, \(G\) and \(G'\), on the enlarged set as follows. Between two classes corresponding to vertices joined in \(R\) draw all edges for \(G\), and put an (\(\varepsilon, \delta\))-super-regular bipartite graph for \(G'\). If \(H\), a graph of degree at most \(\Delta\) is embeddable into \(G\) then it is embeddable already into \(G'\).